微分代数包含式の摩擦と接触のモデル化への応用...ALGEBRAIC INCLUSIONS TO THE...

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微分代数包含式の摩擦と接触のモデル化への応用

熊, 小剛

https://doi.org/10.15017/1441188

出版情報：Kyushu University, 2013, 博士（工学）, 課程博士バージョン：権利関係：Fulltext available.

APPLICATIONS OF DIFFERENTIAL

ALGEBRAIC INCLUSIONS TO THE

MODELING OF FRICTION AND CONTACT

by

XIAOGANG XIONG

Acknowledgments

The writing of this dissertation would have never been possible without the financial sup-

port provided by the scholarships from the government of the P. R. of China and Kyushu

University in Japan, and emotional support from many people during my Ph.D. research at

Kyushu University.

I first would like to sincerely thank my chief supervisor, Professor Motoji Yamamoto, for

providing me lots of help during my study in Kyushu University. He gave me the opportunity

and courage to study in Japan. His kindness, encouragement, and patience have been great

support throughout my research.

I would like to particularly thank my co-supervisor, Associate Professor Ryo Kikuuwe,

for his enthusiastic and patient academic guidance. Any progression I made on my academic

research cannot happen without his direction, education, and inspiration. I have benefited a

lot from his excitation to science and engineering research.

I wish to express my thanks to Professor Takahiro Kondou, Professor Joichi Sugimura,

Professor Svinin Mikhail, Associate Professor Kenji Tahara, Assistant Professor Takeshi

Ikeda for their suggestions and comments throughout my Ph.D. research. I am also grateful

to Mr. Kouki Shinya and Ms. Midori Tateyama for their help in various forms. Furthermore,

I want to thank all previous and current Ph.D. and master students of Control Engineering

Laboratory, Dr. Katsuki , Dr. Jin, Mr. Aung, Mr. Iwatani, and so on, for various interesting

talks and discussions.

Finally, I am deeply thankful to my parents for their moral support and encouragement

throughout my Ph.D. research.

I

Contents

List of figures V

List of abbreviations XIII

1 Introduction 1

1.1 Mechanical systems involving friction and contact . . . . . . . . . . . . . . 1

1.2 Differential inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Friction modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Contact modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Major achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Mathematical preliminaries 13

2.1 Signum function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Diode function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 General notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Mechanical systems involving Coulomb friction and rigid unilateral contact 17

3.1 Previous simulation approaches . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Coulomb friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.2 Rigid unilateral contact . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 New simulation approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.1 Coulomb friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.2 Rigid unilateral contact . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.3 Coulomb-frictional, rigid unilateral contact . . . . . . . . . . . . . 27

II

CONTENTS III

3.2.4 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Examples and simulation results . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Example I: A rolling sphere with collision and slip . . . . . . . . . 30

3.3.2 Example II: Multiple friction with multiple rigid unilateral contacts 33

3.3.3 Example III: Periodic motion . . . . . . . . . . . . . . . . . . . . . 37

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 A friction model with realistic presliding behavior 41

4.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.1 LuGre model, single-state elastoplastic model, Leuven model and

modified Leuven model . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.2 Generalized Maxwell-Slip friction model . . . . . . . . . . . . . . 43

4.1.3 Differential-algebraic single-state friction model . . . . . . . . . . 44

4.2 New multistate friction model . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2.1 Extension to include Stribeck effect . . . . . . . . . . . . . . . . . 46

4.2.2 Extension to include presliding hysteresis with nonlocal memory . . 47

4.2.3 Extension to include frictional lag . . . . . . . . . . . . . . . . . . 48

4.3 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 Simulation and comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4.1 Nondrifting and stick-slip oscillation . . . . . . . . . . . . . . . . 51

4.4.2 Rate independency and amplitude dependency of hysteresis loop . . 53

4.4.3 Frictional lag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4.4 Comparison between GMS model and new model . . . . . . . . . . 57

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 A contact model with nonlinear compliance 65

5.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 New contact model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 Formulation of new model . . . . . . . . . . . . . . . . . . . . . . 69

5.2.2 Comparison to HC model . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Effects of parameters β1, β2 and γ . . . . . . . . . . . . . . . . . . . . . . 72

Contents IV

5.3.1 Force-indentation curves . . . . . . . . . . . . . . . . . . . . . . . 72

5.3.2 Coefficient of restitution (COR) . . . . . . . . . . . . . . . . . . . 79

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Concluding remarks 81

References 84

Appendix 95

List of figures

1.1 (a) One of typical ways to model friction in hard-constraints approaches,

and (b) one of typical ways in regularization approaches . . . . . . . . . . 4

1.2 (a) One of typical ways to model contact in hard-constraints approaches,

and (b) one of typical ways in regularization approaches . . . . . . . . . . 9

2.1 The signum function sgn(x) and saturation function sat(Z, x) . . . . . . . 14

2.2 The diode function dio(x) and maximum functionmax(0,−x) . . . . . . . 15

2.3 The normal cone of the set S at O1, O2, O3, and O4. . . . . . . . . . . . . . 16

3.1 One physical interpretation of (3.1) . . . . . . . . . . . . . . . . . . . . . . 18

3.2 One physical interpretation of (3.4) . . . . . . . . . . . . . . . . . . . . . . 20

3.3 A physical interpretation of (3.8). . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Simulation of the system (3.1); (a) provided external force fe described as

(3.11); (b) simulation results by RK4 with the timestep size 0.001s. The

parameters are chosen as; M = 1 kg, F = 0.5 N, K = 5 × 103 N/m,

β = 2 × 10−3 s. The initial conditions are; p = 0 m, p = 0 m/s. . . . . . . 23

3.5 A physical interpretation of (3.12) . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Simulation of the system (3.16) by RK4 with the timestep size 0.001s. The

parameters are chosen as; M = 1 kg, fe = −9.8 N, K = 105 N/m, β =

0.01 s, α = 0.01 s(gray dashed), 0.007 s(black dashed), 0.005 s(gray solid),

0.001 s(black solid). The initial conditions are; p = 1 m, p = 0 m/s. . . . . 26

V

List of figures VI

3.7 Example I: A rolling sphere with collision and slip. In the simulation, the

parameters are chosen as; M = 1 kg, R = 0.5 m, µ = 0.1 and the initial

conditions are; p = [5.5, 0, 0]T m/s, p = [0, 0, 2R]T m, ω = [0, 0, 0]T rad/s. 30

3.8 Simulation results of Example I by using (3.27) integrated by RK4 with

the timestep size 0.001 s. The parameters are chosen as; K1 = K2 =

1 × 105 N/m, β1 = β2 = 4 × 10−3 s, α = 2.8 × 10−3 s. . . . . . . . . . . . 31

3.9 Example II: Multiple friction contacts with multiple rigid unilateral con-

tacts. In the simulation, the parameters were chosen as; µ1 = µ2 = µ3 =

0.5, M1 = 0.5 kg, M2 = 1 kg, Ks = 100 N/m, u = 1 m/s and the initial

conditions are; p = [0, 0.25, 0.05, 0]T m, p = [0, 0, 0, 0]T m/s. . . . . . . . 34

3.10 Simulation results of Example II by using (3.35) integrated by RK4 with

timestep size 0.001s. The gray regions indicate the time periods in which

the objects M1 and M2 are in contact with each other. The parameters are

chosen as; Ki = 5 × 105 N/m, βi = 2 × 10−3 s (∀i ∈ 1, · · · , 6), αi =

1.6 × 10−3 s (∀i ∈ 1, 2, 3). . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.11 Example III: Periodic motion. In the simulation, the parameters were chosen

as; F = 0.2 N, M = 1 kg, k1 = 1 N/m, k2 = 1 N/m3, u = 1 m/s,

V = 1 m/s, and the initial conditions are adopted from [1] for comparison;

p = 1.19149 m, w = 0 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.12 Simulation results of Example III by using (3.40) and (3.41) integrated by

RK4 with timestep size 0.001 s. The parameters in (3.40) are chosen as;

K = 1 × 105 N/m, β = 0.5 s. The parameter in (3.41) is chosen as ϵ =

10−5 m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 A physical interpretation of (4.3b) and (4.3b). In the sliding regime, the

asperity deflections ai evolve so that the elastic forces |κiai| converge to

λig(v). The total friction force f is the sum of the viscoelastic forces of the

elements. Figure and caption are reprinted, with permission from Springer:

Tribology letter, “A Multistate Friction Model Described by Continuous

Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong,

Ryo Kikuuwe, and Motoji Yamamoto, Fig. 2 (see Appendix A1). . . . . . . 43

List of figures VII

4.2 A physical interpretation of the model (4.4a) and (4.4b). The viscoelas-

tic force balances the friction force FcSgn(v − a) both in the sliding and

presliding regimes. Figure and caption are reprinted, with permission from

Springer: Tribology letter, “A Multistate Friction Model Described by Con-

tinuous Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013, Xiaogang

Xiong, Ryo Kikuuwe, and Motoji Yamamoto, Fig. 3 (see Appendix A1). . . 45

4.3 A physical interpretation of the model (4.10a) and (4.10b). Figure and

caption are reprinted, with permission from Springer: Tribology letter, “A

Multistate Friction Model Described by Continuous Differential Equations”,

vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong, Ryo Kikuuwe, and Mo-

toji Yamamoto, Fig. 4 (see Appendix A1). . . . . . . . . . . . . . . . . . . 49

4.4 Nondrifting property; (a) an external force as a function of time t; (b) the

friction force f as a function of position p. The parameters are chosen

as; α = 2, vs = 5 µm/s, Fc = 1 N, Fs = 2.5 N, κi = (5/i) N/µm,

σi = 0.02 Ns/µm, λi = 0.1, i ∈ 1, 2, · · · , 10, τd = 0.02 s. The time-step

size was 0.001 s. Figure and caption are reprinted, with permission from




4.5 The scenario of stick-slip oscillation . . . . . . . . . . . . . . . . . . . . . 53

4.6 Stick-slip oscillation; (a) the position and velocity of the mass M as a func-

tion of time t; (b) the spring force and friction force as a function of time

t. The parameters were chosen identical to those in Fig. 4.4. The time-step

size was 0.005 s. Figure and caption are reprinted, with permission from




List of figures VIII

4.7 Rate independency of hysteresis loop; (a) two positional inputs p as func-

tions of time t (The gray curve is of 10 times the frequency of the black

curve.); (b) the friction force f obtained for the two different input position

signals p. The parameters were chosen identical to those in Fig. 4.4. The

time-step size was 0.01 s. Figure and caption are reprinted, with permission

from Springer: Tribology letter, “A Multistate Friction Model Described by

Continuous Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013, Xi-

aogang Xiong, Ryo Kikuuwe, and Motoji Yamamoto, Fig. 6 (see Appendix

A1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.8 Amplitude dependency of hysteresis loop; (a) four positional inputs p with

different amplitudes as functions of time t; (b) the friction forces f obtained

for the four different input position signals p. The parameters were cho-

sen identical to those in Fig. 4.4. The time-step size was 0.001 s. Figure

and caption are reprinted, with permission from Springer: Tribology letter,

“A Multistate Friction Model Described by Continuous Differential Equa-

tions”, vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong, Ryo Kikuuwe,

and Motoji Yamamoto, Fig. 7 (see Appendix A1). . . . . . . . . . . . . . . 60

4.9 Frictional lag; (a) two velocity inputs v as functions of time t; (b) the friction

force f obtained for the two different input velocity signals v and two values

of τd. The other parameters were chosen identical to those in Fig. 4.4. The

time-step size was 0.001 s. Figure and caption are reprinted, with permission

from Springer: Tribology letter, “A Multistate Friction Model Described by

Continuous Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013, Xi-

aogang Xiong, Ryo Kikuuwe, and Motoji Yamamoto, Fig. 8 (see Appendix

A1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

List of figures IX

4.10 Frictional lag effect on transition behavior; (a) two velocity input v as func-

tions of time t; (b) the friction force f obtained for the two different input

velocity signals v and two values of τd. The other parameters were chosen

identical to those in Fig. 4.4. The time-step was 0.001 s. Figure and caption

are reprinted, with permission from Springer: Tribology letter, “A Multistate

Friction Model Described by Continuous Differential Equations”, vol. 51,

no. 3, pp. 513–523, 2013, Xiaogang Xiong, Ryo Kikuuwe, and Motoji Ya-

mamoto, Fig. 9 (see Appendix A1). . . . . . . . . . . . . . . . . . . . . . 62

4.11 Transition behaviors of the GMS model (4.3b)(4.3b) and new model (4.13a)

(4.13b) (4.13c); (a) a velocity input v as a functions of time t; (b)(c)(d) the

friction force f obtained from the input velocity v; (e)(f)(g)(h) the friction

force f and the number of elements in the presliding regimes; The gray ver-

tical lines in (e)(f)(g)(h) indicate the time at which the number of elements

in the presliding regime changes. The parameters were chosen identical to

those in Fig. 4.4. The time-step size was 0.0001 s. (It should be noted that

non-zero σi has not been used in reported applications of the GMS model

in the literature.) Figure and caption are reprinted, with permission from



Xiong, Ryo Kikuuwe, and Motoji Yamamoto, Fig. 10 (see Appendix A1). . 63

4.11 (Continued.) Figure and caption are reprinted, with permission from Springer:

Tribology letter, “A Multistate Friction Model Described by Continuous

Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong,

Ryo Kikuuwe, and Motoji Yamamoto, Fig. 10 (see Appendix A1). . . . . . 64

List of figures X

5.1 Force-indentation curves; (a) typical empirical result adopted from, e.g.,

[2, Fig. 4], [3, Fig. 2.6] and [4, Fig. 2], (b) KV model (5.1), (c) HC model

(5.2) without any external force (solid) and with an external pulling force

(dashed), (d) the author’s previous contact model (5.3). Figure and caption

are reprinted, with permission from ASME: Journal of Applied Mechanics,


tions”, vol. 81, no. 2, pp. 021003-1:8, 2014, Xiaogang Xiong, Ryo Kikuuwe,


5.2 The force-indentation curves of the new model (5.6) and HC model (5.2);

The parameters are chosen as; fe = 0 N, λ = 1.5, K = 104 N/m1.5,

γ = 2×103 s−1, β1 = 3×10−3 s, β2 = 0.1 s/m1.5 and b2 = 0.35 s/m. The ini-

tial conditions are set as; p(0) = −0.1 m, p(0) = 2 m/s and a(0) = 0 m1.5.

Figure and caption are reprinted, with permission from ASME: Journal of

Applied Mechanics, “A Multistate Friction Model Described by Continu-

ous Differential Equations”, vol. 81, no. 2, pp. 021003-1:8, 2014, Xiaogang


5.3 Simulation of bouncing motion by using the new model (5.6) and HC model

(5.2). The parameters are set as; fe = Mg, λ = 1.5, K = 104 N/m1.5, γ =

2× 103 s−1, β1 = 1.45× 10−3 s, β2 = 0.2 s/m1.5 and b2 = 0.2 s/m. The ini-

tial conditions are set as; p(0) = −0.5 m, p(0) = 0 m/s and a(0) = 0 m1.5.

Figure and caption are reprinted, with permission from ASME: Journal of

Applied Mechanics, “A Multistate Friction Model Described by Continu-

ous Differential Equations”, vol. 81, no. 2, pp. 021003-1:8, 2014, Xiaogang


List of figures XI

5.4 Influence of β1 on the behaviors of the new model (5.5); The parameters are

set as; fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The initial conditions are set

as; p(0) = −0.1 m, p(0) = 2 m/s and a(0) = 0 m1.5. Figure and caption are

reprinted, with permission from ASME: Journal of Applied Mechanics, “A


vol. 81, no. 2, pp. 021003-1:8, 2014, Xiaogang Xiong, Ryo Kikuuwe, and

Motoji Yamamoto, Fig. 5 (see Appendix A2). . . . . . . . . . . . . . . . . 73

5.5 Influence of β2 on the behaviors of the new model (5.5); The parameters are

set as; fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The initial conditions are set

as; p(0) = −0.1 m, p(0) = 2 m/s and a(0) = 0 m1.5. Figure and caption are

reprinted, with permission from ASME: Journal of Applied Mechanics, “A


vol. 81, no. 2, pp. 021003-1:8, 2014, Xiaogang Xiong, Ryo Kikuuwe, and

Motoji Yamamoto, Fig. 6 (see Appendix A2). . . . . . . . . . . . . . . . . 74

5.6 Influence of γ on the behaviors of the new model (5.5); The parameters are

set as; fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The initial conditions are

set as; p(0) = −0.1 m and a(0) = 0 m1.5. Figure and caption are reprinted,

with permission from ASME: Journal of Applied Mechanics, “A Multistate

Friction Model Described by Continuous Differential Equations”, vol. 81,

no. 2, pp. 021003-1:8, 2014, Xiaogang Xiong, Ryo Kikuuwe, and Motoji

Yamamoto, Fig. 7 (see Appendix A2). . . . . . . . . . . . . . . . . . . . . 75

5.7 (a)(b) The COR as a function of β1, β2 and the impact velocity obtained

from of the new model (5.5). (c)(d) The COR as a function of impact veloc-

ity. The parameters are set as; fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The

initial conditions are set as; p(0) = −0.1 m and a(0) = 0 m1.5. Figure and

caption are reprinted, with permission from ASME: Journal of Applied Me-

chanics, “A Multistate Friction Model Described by Continuous Differen-

tial Equations”, vol. 81, no. 2, pp. 021003-1:8, 2014, Xiaogang Xiong, Ryo

Kikuuwe, and Motoji Yamamoto, Fig. 8 (see Appendix A2). . . . . . . . . 77

List of figures XII

5.8 (a) The COR as a function of γ and the impact velocity obtained from of

the new model (5.5). (b) The COR as a function of impact velocity. The

parameters are set as; fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The initial

conditions are set as; p(0) = −0.1 m and a(0) = 0 m1.5. Figure and caption

are reprinted, with permission from ASME: Journal of Applied Mechanics,


tions”, vol. 81, no. 2, pp. 021003-1:8, 2014, Xiaogang Xiong, Ryo Kikuuwe,


List of abbreviations

DI Differential inclusion

DAI Differential algebraic inclusion

HC Hunt-Crossley

GMS Generalized Maxwell-slip

COR Coefficient of restitution

ODE Ordinary differential equation

PID Proportional integral derivative

XIII

Chapter 1

Introduction

1.1 Mechanical systems involving friction and contact

Friction and contact are common phenomena that occur in almost all mechanical systems.

The two phenomena seems simple, but in fact they are much complicated, involving many

factors, such as temperature and materials. Although it is impossible to give exact mathe-

matical descriptions for friction and contact, they can be modeled by different simplified

ways in different cases. According to the simplifications of friction and contact models,

this dissertation classifies the description methods of mechanical systems into two groups,

hard-constraint approaches and regularization approaches.

In some cases, the deformations caused by the friction and contact phenomena are neg-

ligible, comparing to the volume sizes of contact bodies. In those cases, the contact bodies

can be idealized as rigid ones and can be assumed to be impenetrable to each other. From

a mathematical point of view, friction and contact phenomena are described as tangential

and normal constraints, respectively. This kind of descriptions for mechanical systems in-

volving friction and contact are termed as hard-constraint approaches. They are preferred

by academic researchers in the filed of multibody dynamical systems, because those hard-

constraint approaches allow for great simplifications and lead to more robust numerical

schemes for simulations of mechanical systems with friction and contact. Typical exam-

ples of mechanical systems described by hard-constraint approaches include woodpecker

1

Chapter 1. Introduction 2

toy [5, 6], slider-crank mechanism with a translational clearance joint [7, 8], and granular

material with large quantity [9–11].

In some other cases, friction and contact are modeled by considering the microscopic

deformations from the physical point of view, as opposed to the mathematical point of view.

These models, usually consisting of spring and damper elements, are regularizations of the

hard-constraint approaches. Those kind of description methods for mechanical systems are

therefore termed as regularization approaches. They can be viewed as more realistic and

closer to real systems than the hard-constraint approaches, because the variables and param-

eters in regularization approaches represent specific physical meanings, such as displace-

ments, stiffness, and viscosity. Regularization approaches are appreciated by engineers in

the field of virtual reality, especially, in cases where the contact bodies are not so rigid or

even soft [12]. Regularized models of friction and contact are also applicable in mechanical

engineering, such as friction compensations with regularized friction models [13, 14] and

property estimations of soft object with regularized contact models [2–4, 15].

The friction and contact models in regularization approaches can be continuously ex-

tended to more sophistical models based on observations of experimental data. For example,

the Hertz contact model is one of contact models in regularization approaches. It has been

extended into Kelvin-Voigt model, Hunt-Crossly model [16], and other more complicated

contact models [17].

1.2 Differential inclusion

Hard-constraint approaches describe mechanical systems involving friction and contact as

differential inclusions (DIs), which are generalizations of ordinary differential equations

(ODEs) as the following form:

d

dtx(t) ∈ F (x(t), u(t)) (1.1)

where x(t) is a vector that represents the states of systems, u(t) is another vector repre-

senting the inputs, and F (·, ·) is a set-valued map rather than a single-valued one. The


map F (x(t), u(t)) is only set-valued at countable points of x and is single-valued at other

values of x. The DI description (1.1) has the difficulty in numerical integration due to its

set-valued characteristic. For example, DIs in [7, 18] have to be discretized by some Euler-

like methods and then integrated numerically by special mathematical solvers, such as linear

complementary problem (LCP) solvers and mixed linear complementary (MLCP) solvers.

In some cases, DIs cannot be explicitly written in the form (1.1). They can only be

written in the form of differential algebraic inclusions (DAIs) as follows:

0 ∈ F (x(t), x(t), u(t)). (1.2)

In the main content of this dissertation, i.e., Chapter 3, 4, and 5, DIs in the form of

(1.1) is regularized into DAIs (1.2) by considering the microscopic deformations caused by

friction and contact. This way provides a new perspective for DI regularizations, which is

different from the conventional regularization approaches.

1.3 Friction modeling

Hard-constraint approaches typically describe the friction force fT ∈ R as a set-valued

function of the velocity vT ∈ R as follows [18, 19]:

fT ∈

−F if vT > 0

[−F, F ] if vT = 0

F if vT < 0

(1.3)

where F is the magnitude of kinetic friction force. The friction force at the static fric-

tion state is set-valued within the range [−F, F ], as shown by the vertical line segment

in Fig 1.1(a). Such kind of discontinuous characteristic is important to describe the static

friction [14]. It is easy to imagine that the set-valued friction model (1.3) leads to DI de-

scriptions for mechanical systems involving friction.

In regularization approaches, the friction force fT is typically described as a continuous


vT0

F

¡F

fT

vT0

F

¡F

fT

V¡V

vT

M

(a)

vT

M

(b)

fT

fT

Figure 1.1: (a) One of typical ways to model friction in hard-constraints approaches, and (b)

one of typical ways in regularization approaches


function of the velocity vT as follows:

fT =

−F if vT > V

−FvT /V if |vT | ≤ V

F if vT < −V

(1.4)

where V > 0 is a threshold that is used to replace zero value for easinesses of numerical in-

tegration and F/V > 0 can be interpreted as the viscosity coefficient. This friction force law

is illustrated in Fig 1.1(b). One can notice that the vertical portion in Fig 1.1(a) is replaced

by a line segment with slope −F/V . This implies that the friction force is a continuous

function of the velocity. It is easy to imagine that such kind of friction model lead to ODE

descriptions for mechanical systems involving friction, without any integration problems.

However, due to the lack of the discontinuous nature, such kind of friction models cause

various strange behaviors. For example, the behavior of the friction model (1.4) heavily

depends on the chosen value of the threshold V . Some more elaborately regularized models

than (1.4), e.g., Dahl [20] and LuGre [21] friction models, produce positional drift in the

static friction state.

Experimental data of friction in the literature, e.g., [22, 23], show that friction phenom-

ena are more complicated than what is described by Fig 1.1(a) and Fig 1.1(b). From static

friction to kinetic friction, friction is believed to experience two different regimes: the pres-

liding regime and the sliding regime [24]. The presliding regime is the time period within

which the major asperities in the contact surfaces are deformed elastically. The friction force

is a function of presliding displacement, which is a consequence of the elastic deformations

of asperities. In contrast, the sliding regime is the time period within which the major asper-

ities are deformed inelastically and the interconnections of asperities between the contact

surfaces are broken. The friction force is a function of the velocity, as roughly described

by Coulomb friction model. It has been pointed out [24, 25] that, in reality, the transitions

between the presliding and sliding regimes are not always sudden but gradual, and some

features are observed in each regime, such as the presliding hysteresis with nonlocal mem-


ory1 in the presliding regime and frictional lag2 in the sliding regime. Smooth transitions

between the two regimes and the features in each regime should be appropriately described

by friction models for applications such as precise control of machines [27, 28] and precise

simulations of mechanical systems [29, 30].

The Dahl model [20], the LuGre model [21, 31], and the single state elastoplastic model

[32] are relatively simple models that are preferred for control applications. Each of them

has one internal state variable representing the average deflection of asperities in the contact

surfaces, and thus they can be referred to as single-state friction models. As a price for

their simplicity, these three models do not capture some important properties of friction

phenomena in the presliding regime. For instance, the Dahl and LuGre models produce

unbounded positional drift even when the applied force is smaller than the maximum static

friction force [32, 33]. The single state elastoplastic model is free from this problem, but

none of those three models produce presliding hysteresis with nonlocal memory.

The modified Leuven model [23, 34], the generalized Maxwell-slip (GMS) model [35],

and the smoothed GMS model [36, 37] can be referred to as multistate friction models be-

cause each of them includes more than one internal state variables. Those three multistate

models are free from the aforementioned problems of the single-state friction models due

to their sophisticated structures, but they still suffer from other problems. The modified

Leuven model and the GMS models are described by equations involving discontinuities re-

sulting from the switching between the presliding and sliding regimes. Such switching struc-

tures cause the difficulty in on-line parameters identification [36, 37]. The smoothed GMS

model does not involve such discontinuities, but comparing to the original GMS model, this

smoothed version additionally includes two functions and three parameters, which increases

the difficulty in parameter identification. The formulations of the smoothed GMS model are

any order smoothly differentiable. This characteristic is originally motivated from the easi-

ness of parameter identifications, without any physical meanings.

1Hysteresis with nonlocal memory is defined as an input-output map, of which the output at any time

instant depends on the output at some time instant in the past, the input since then, and the past extrema of the

input or the output [22, 24]. This is in contrast to hysteresis with only local memory, of which the output only

depends on the output at some time instant in the past [24].2Frictional lag is time delay in the friction force as a function of velocity. It results in the friction force

being larger during acceleration than that during deceleration [14, 21, 26].


There are some physically-motivated models that consider microscopic dynamics of

contact surfaces. For example, generic models [38, 39] consider the effects of masses and

geometries of asperities in the contact surfaces. As pointed out in [23, 40], the generic mod-

els give good agreements with experimental data, but they can be too complex for control

applications due to the necessity of massive computation. The Burridge-Knopoff [41] and

its modified versions, such as [42], also consider the masses of asperities and the stiffness

of the couplings among the asperities. Those models can also be computationally expensive

compared to the LuGre, Leuven, and GMS models, which are suited for control applications.

In Chapter 3, a single-state friction model derived from a DAI is used to approximate

the hard-constraint model (1.3). Unlike the regularized model (1.4), the Dahl model, and the

LuGre model, this single-state model is independent from any velocity thresholds and does

not produce unbounded positional drift, but it produces only linearly elastic behavior in the

presliding regime. Chapter 4 extends the DAI proposed in Chapter 3 to a multistate version

with including Stribeck and frictional lag effects. This multistate version is composed of

parallel connection of a number of elastoplastic elements, having a similar structure to those

of the conventional multistate models and others [43–46]. The force produced by the ex-

tended model is analytically continuous with respect to the velocity and the state variables

because it is transformable to a set of ODEs with continuous right-hand sides. Moreover, it

captures the Stribeck effect, nondrifting property, stick-slip oscillation, presliding hysteresis

with nonlocal memory, and frictional lag, which are major features of friction phenomena

reported in the literature.

1.4 Contact modeling

Hard-constraint approaches typically describe the contact force fN ∈ R as a set-valued

function of the positional distance pN ∈ R as follows:

fN ∈

0 if pN > 0

[0,∞] if pN = 0

∅ if pN < 0.

(1.5)


The contact force is set-valued within the scope [0, +∞] when the contact is closed, i.e.,

pN = 0, as shown by the vertical portion in Fig. 1.2(a). When the contact is open, i.e.,

pN > 0, the contact force is fN = 0. The empty set at pN < 0 implies that the contact

bodies are idealized as rigid and impenetrable. The nonnegative force fN ≥ 0 in (1.5)

implies that the contact bodies are not sticky to each other. One can notice the orthogonal

corner at the origin in Fig 1.2(a), which shows the discontinuities of contact model (1.5).

It is easy to imagine that such a set-valued model leads to DI descriptions for mechanical

systems involving contact.

In regularization approaches, the relation between fN and pN is typically described as a

continuous function fN = max(0,−kpN) where k can be interpreted as the stiffness. One

can notice that the vertical portion in Fig 1.1(a) has been replaced by an inclined one with

slope −k in Fig 1.2(b). The contact force is zero when the contact is open, i.e., pN > 0. The

contact force is proportional to the penetration −pN when the contact is closed, pN ≤ 0.

The Hertz contact model can be viewed as a nonlinear extension of this regularized model.

The contact between two bodies is in fact more complicated than that described by the

hard-constraint and regularization approaches in Fig 1.2(a) and (b). The force exerted by

a contact varies according to nonlinear relations with time and the relative displacement,

both during the contact and at the beginning and end of the contact. In order to precisely

simulate the behaviors of various mechanical systems, such relations must be appropriately

modeled. Much effort has been paid for the modeling of contact, especially those among

biological tissues [2, 3], components of machinery [47, 48], rubber materials [4, 15, 49, 50]

and granular materials [51, 52].

The Kelvin-Voigt (KV) model is one of simply regularized models. It is based on a linear

spring-dashpot model and it captures energy dissipation during contact [53]. The drawback

of this model is that it produces a discontinuous jump in the force at the beginning of contact

and a sticky force (negative contact force) in the end of contact [16, 53, 54]. Hunt-Crossley

(HC) model [16] is another compliant contact model that is free from such weaknesses. HC

model is a combination of Hertz-like nonlinear compliance and an indentation-dependent

damping term. This model is extended in [12, 54–57] and empirically validated in [2, 3,

50, 58–60]. As pointed out in [55], however, HC model can produce unnatural sticky force


0

0

(a)

(b)

fN

pN

fN

pN

vN

pN>0

M

vN

pN>0

M

Figure 1.2: (a) One of typical ways to model contact in hard-constraints approaches, and (b)

one of typical ways in regularization approaches


especially when the contact objects are separated rapidly by an external force. Moreover,

the experiments in [60] show that the HC model is only consistent with empirical results

when the coefficient of restitution (COR) is large [61].

The literature includes some report regarding the force-indentation curves obtained from

real objects [2–4, 15, 49, 60]. Those force-indentation curves agree with the curves of HC

model in most aspects, but differ in that, when the force arrives back in zero, the indentation

is still not zero. That is, one can say that a contact force model should satisfy the following

three conditions:

(i) the contact force should be continuous with respect to time at the time of contact;

(ii) the force-indentation curve can be designed to be nonlinear as is in Hertz’s contact

model;

(iii) the indentation can be nonzero at the time of loss of contact force.

As discussed earlier, KV and HC models fail to satisfy two and one, respectively, of the three

conditions. There have been some models [49, 61–64] that satisfy those conditions. These

models, however, describes the restitution phase and the compression phase separately, in

different equations from each other. As far as the author are aware, there have been no

contact model that realizes the aforementioned features in a unified framework. In this

dissertation, lacking of discontinuity nature is thought to be the reason for their problems of

those models.

In Chapter 3, a linear contact model derived from a DAI is used to approximate the

hard-constraint model (1.5). This model does satisfy the conditions (i) and (iii) but does not

satisfy the condition (ii), producing a linear force-indentation curve. Chapter 5 extends the

DAI proposed in Chapter 3 to a new nonlinear DAI by adding a Hertz-like power-law non-

linearity and displacement-dependent viscosity such that the equivalent ODE formulation,

i.e., the new contact model, satisfies the aforementioned three conditions.


1.5 Major achievements

This dissertation focuses on modeling friction and contact for simulation and engineering

applications. First, this dissertation regularizes DIs by using DAIs for simulations of me-

chanical systems involving friction and contact. In contrast to previous regularization ap-

proaches, the new approach preserve the discontinuity of original DIs. It is free from various

problems of unnatural behaviors produced by the conventional regularization approaches.

Then, this dissertation extends the DAIs of friction and contact to more sophisticated formu-

lations for engineering applications. Extensive simulation studies show that the properties

of the new models derived from the extended DAIs are consistent with the experimental data

reported in the literature. The major achievement are as follows:

• New simulation method for mechanical systems (Chapter 3)

This method provides a new perspective to regularize DIs for simulations of me-

chanical systems. Different from straightforward approximations of DIs by ODEs in

conventional regularization approaches, this new model first relaxes the rigid-bodies

based DIs into DAIs with considering the deformations of bodies. Then, those DAIs

are equivalently transformed into ODEs. Therefore, it has the features of both hard-

constraint and regularization approaches and possesses both of their advantages. The

new approximation method makes it feasible to simulate complicated mechanical sys-

tems with an easy integration procedure as regularization approaches. On the other

hand, it preserves the discontinuous nature of the original systems.

• New friction model (Chapter 4)

This new friction model is an extended version of the friction model proposed in

Chapter 3. This extended model is free of the positional drift problem of Dahl and

LuGre model, unnatural nondrifting behavior of Leuven model, and discontinuous

force of the GMS model. The problems of first three models are caused by lacking

of the discontinuous nature of friction between presliding and sliding regimes. The

GMS model does not suffer from the problems of the previous three models due to its

preservation of discontinuous nature, but it encounters the mathematical difficulty in

dealing with transitions between the presliding and sliding regimes. A pair of switch-


ing conditions have to be used in the GMS model for the transitions. On the contrary,

the extended friction model not only preserves the discontinuous nature of friction,

but also allows smooth transitions between the presliding and sliding regimes with-

out any switching structures. In addition, the extended friction model captures all the

properties of friction that the GMS model does.

• New contact model (Chapter 5)

This new model is a nonlinear extension of the contact model proposed in Chapter 3. It

can simultaneously satisfy those three features of contact phenomena observed from

experimental data without suffering from the problems of the KV and HC models.

Different from the KV and HC models, this new model has two viscosity terms for

energy dissipation, which leads to adjustable coefficient of restitutions (CORs) even at

low impact velocities. Those two terms have similar physical meanings with respect

to those in the KV and HC models. Furthermore, it can produce similar behaviors to

the KV and HC models by adjusting the two viscosity terms. Therefore, both the KV

model and HC model can be viewed as special cases of this extended contact model

in a unified form.

1.6 Organization

The rest of this dissertation is organized as follows. Chapter 2 provides mathematical pre-

liminaries that will be used throughout this dissertation. Chapter 3 proposes to regularize

DIs into DAIs for simulations of mechanical systems involving friction and contact. It illus-

trates the simplicity and efficiency of this new approach through three examples. Chapter 4

extends the single-state friction model in Chapter 3 to a multistate version and illustrates

the consistences between the extended model and experimental data reported in the litera-

ture. Chapter 5 extends the linear contact model in Chapter 3 to a new nonlinear version

and illustrates its properties through numerical simulations. Chapter 6 provides concluding

remarks.

Chapter 2

Mathematical preliminaries

For the discussion throughout this dissertation, this section introduces three functions: sgn,

sat and dio. Some theorems regarding those functions are also presented. In the rest of this

dissertation, R denotes the set of all real numbers and R+ denotes the set of all nonnegative

real numbers. The symbol 0 denotes the zero vector of an appropriate dimension.

2.1 Signum function

First, let us define the signum function sgn : Rn → R

n and the unit saturation function

sat : R+ × Rn → R

n as follows:

sgn(x)∆=

x/∥x∥ if ∥x∥ = 0

z ∈ Rn | ∥z∥ ≤ 1 if ∥x∥ = 0

(2.1)

sat(Z, x)∆=

Zx/∥x∥ if ∥x∥ > Z

x if ∥x∥ ≤ Z(2.2)

where x ∈ Rn, Z ∈ R+ and ∥ · ∥ denotes the vector two-norm. If n = 1, the sgn(x) and

sat(Z, x) can be depicted as Fig. 2.1(a) and Fig. 2.1(b), respectively. The following theorem

is useful to rewrite the DIs involving sgn as ODEs involving sat:

13

Chapter 2. Mathematical preliminaries 14

x

sgn(x)

0

1

¡1

x

sat(Z, x)

0

Z

¡Z

(a) (b)

Z

¡Z

Figure 2.1: The signum function sgn(x) and saturation function sat(Z, x)

Theorem 1. For x, y ∈ Rn and Z ∈ R+, the following relation holds true [65, 66]:

y ∈ Zsgn(x − y) ⇔ y = sat(Z, x). (2.3)

Proof. : A proof can be given as follows:

y ∈ Zsgn(x − y) ⇔ (y = Z(x − y)/∥x − y∥ ∧ x = y) ∨ (y = x ∧ ∥y∥ ≤ Z)

⇔ (y = Zx/(Z + ∥x − y∥) ∧ x = y ∧ ∥y∥ = Z) ∨ (y = x ∧ ∥x∥ ≤ Z)

⇔ (y = Zx/(Z + ∥x − y∥) ∧ ∥x∥ = Z + ∥x − y∥ > Z) ∨ (y = x ∧ ∥x∥ ≤ Z)

⇔ (y = Zx/∥x∥ ∧ ∥x∥ > Z) ∨ (y = x ∧ ∥x∥ ≤ Z) ⇔ y = sat(Z, x).

2.2 Diode function

Next, let us define the “diode” function dio : R+ → R+ as follows:

dio(x)∆=

0 if x > 0

R+ if x = 0(2.4)


dio(x)

0

x x

max(0, ¡x)

0

(c) (d)

1

¡1

Figure 2.2: The diode function dio(x) and maximum functionmax(0,−x)

where x ∈ R+. The following theorem is useful to rewrite DIs involving dio by ODEs:

Theorem 2. For x ∈ R and y ∈ R+, the following relation holds true:

y ∈ dio(x + y) ⇔ y = max(0,−x). (2.5)

Proof. : A proof can be given as follows:

y ∈ dio(x + y) ⇔ (y = 0 ∧ x + y > 0) ∨ (y ≥ 0 ∧ x + y = 0)

⇔ (y = 0 ∧ x > 0) ∨ (y ≥ 0 ∧ y = −x) ⇔ y = max(0,−x).

The graphs of dio(x) and max(0,−x) are illustrated as Fig. 2.2(a) and Fig. 2.2(b), re-

spectively.

2.3 General notations

It must be noted that Theorem 1 and Theorem 2 are special cases of the following relation,

which has been used in, e.g., [18, Appendix A.3], [19, eq.(2)] and [67, eq.(4)]:

x − y ∈ NS(y) ⇔ y = prox(S, x). (2.6)


O1

O2

O3

O4

S

Figure 2.3: The normal cone of the set S at O1, O2, O3, and O4.

Here, x ∈ Rn, y ∈ S ⊂ R

n, S is a closed convex set, and NS(y) is the normal cone to the

set S at y. The normal cone NS(y) is defined as follows:

NS(y) =

z ∈ Rn|zT (y − ξ) ≥ 0,∀ξ ∈ S, if y ∈ S

∅ otherwise.(2.7)

Examples of two dimensional normal cone NS(y) to the set S at the points O1, O2, O3, and

O4 are illustrated in Fig. 2.3. The normal cones NS(y) at those points are colored with gray.

The function prox(S, x) represents the “proximal point” defined as follows:

prox(S, x)∆= argmin

z∈S

∥z − x∥2. (2.8)

Theorem 1 and Theorem 2 can be obtained by using the relation (2.6) with S = z ∈

Rn | ∥z∥ ≤ Z and S = R+, respectively.

Chapter 3

Mechanical systems involving Coulomb

friction and rigid unilateral contact

Nonsmooth mechanical systems, which are mechanical systems involving Coulomb friction

and rigid unilateral contact, are usually described as DIs. Those DIs may be approximated

by ordinary differential equations (ODEs) by simply smoothing the discontinuities. Such ap-

proximations, however, can produce unrealistic behaviors because the discontinuous natures

of the original DIs are lost. This chapter presents a new algebraic procedure to approximate

DIs describing nonsmooth mechanical systems by ODEs with preserving the discontinu-

ities. The procedure is based on the fact that the DIs can be approximated by differential

algebraic inclusions (DAIs) and thus they can be equivalently rewritten as ODEs. The pro-

cedure is illustrated by some examples of nonsmooth mechanical systems with simulation

results obtained by the fourth-order Runge-Kutta method.

The rest of this chapter is organized as follows. Section 3.1 overviews previous approx-

imation methods for Coulomb friction and rigid unilateral contact. Section 3.2 gives the

main contribution of the work. Section 3.3 provides two example applications of the new

method. Finally, concluding remarks are given in Section 3.4.

⋆ The content of this chapter is partially published in [33].

17

Chapter 3. Mechanical systems involving Coulomb friction and rigid un . . . 18

p

Mfe

f

Figure 3.1: One physical interpretation of (3.1)

3.1 Previous simulation approaches

3.1.1 Coulomb friction

Let us consider the situation where a rigid mass M > 0, of which the position is p ∈ Rr(r ∈

1, 2), is sliding on a fixed surface, as shown in Fig. 3.1. Let us assume that it is subject

to the Coulomb friction force f ∈ R and an external force fe ∈ R. Then, the equation of

motion of the mass is described as the following DI:

Mp = fe − f (3.1)

where

f ∈ F sgn(p) (3.2)

and F > 0 is the magnitude of kinetic friction force1. The direct integration of (3.1)(3.2)

is difficult since the value of sgn(p) is not determined at p = 0, according to the definition

(2.1) of sgn.

Some previous friction models can be viewed as approximations of (3.2). One simple

way is to employ a threshold velocity [13, 68] below which the velocity is considered zero.

This method may be useful to avoid the discontinuity in (3.2) but the non-physical threshold

1Common definitions of Coulomb friction assume that the static friction force can be larger than the kinetic

friction force. This chapter leaves this out of consideration and assumes that the maximum static friction force

is equal to the kinetic friction force.


can produce unrealistic artifacts. Another way is to employ a new state variable which

usually can be interpreted as the displacement of a viscoelastic element. For example, LuGre

friction model [21, 31] without Stribeck effect can be described as follows:

a = p −K∥p∥a

F(3.3a)

f = Ka + B(p − K∥p∥a/F ) + Dp (3.3b)

where a ∈ Rr is the new state variable, K > 0 is a sufficiently large constant and B,D > 0

are constants appropriately chosen to suppress the oscillation in p. Dahl friction model

[20] is a special case of LuGre friction model with D = B = 0. A disadvantage of those

two models is that they produce unbounded positional drift in the static friction state under

oscillatory external force even smaller than the maximum static friction force [69, 70].

Another type of regularized friction models are proposed by Kikuuwe et al. [69, Sec.III.C]

and Bastien and Lamarque [71] based on Backward-Euler method, and by Kikuuwe and Fu-

jimoto [72] based on a modified Runge-Kutta method. A downside of their models is that

they restrict the choice of methods for time integration.

In hard-constraint approaches, the equations of motion are discretized along time by

Euler-like methods. Those discretized equations are usually formulated into complemen-

tarity problems, which are then numerically solved. The literature includes some comple-

mentarity formulations of Coulomb friction in one-dimensional space [8, 73] and in multi-

dimensional space [74–77]. One exception is Kikuuwe et al.’s approach [69, Sec III.A], in

which the discretized equation in a very simple case is analytically solved by the application

of Theorem 2 in the present chapter.

3.1.2 Rigid unilateral contact

Let us consider the one-dimensional system composed of a rigid mass M , of which the

position is p ∈ R, and a fixed rigid wall whose position coincides with the origin, as shown

in Fig. 3.2. The rigid mass is subject to an external force fe ∈ R. Then, the equation of


pM

fef

Figure 3.2: One physical interpretation of (3.4)

motion of the rigid mass is described as the following DI:

Mp = f − fe (3.4)

where

f ∈ dio(p). (3.5)

The integration of (3.4)(3.5) is also difficult due to dio(p), whose value is not determined at

p = 0.

One of trivial methods to approximately realize the contact force f in (3.5) is as follows

[18, 72, 78]:

f =

−Kp − Bp if p ≤ 0

0 if p > 0(3.6)

where K is a sufficiently large positive constant and B is a positive constant large enough

to suppress the oscillation in p. This force law can be viewed as a linear viscoelastic contact

model with the stiffness K and the viscosity B. As pointed out in [79, 80], one drawback of

(3.6) is that it produces an unnatural sucking force toward the wall. This drawback may be


overcome by using the following slightly different one:

f =

max(0,−Kp − Bp) if p ≤ 0

0 if p > 0.(3.7)

However, both (3.6) and (3.7) are discontinuous with respect to p and p. Thus, they are not

suitable for the use with common ODE solvers.

As another example, the nonlinear viscoelastic contact model proposed by Hunt and

Crossley [79] can also be viewed as an approximation of rigid unilateral contact. This

model was extended in [12, 55, 81] and empirically validated in [59, 60, 82]. This model is

continuous with respect to p and p, but it can also produce unnatural sucking force when p

is large.

In hard-constraint approaches for rigid unilateral contact, the equations of motion are

usually discretized by Euler-like methods and then solved numerically [8, 9, 73, 75, 83]. A

different approach is in [18, Sec.1.4.3.2][80] where the discretized equations in very simple

cases are solved analytically. Those methods can be used only with Euler-like methods.

3.2 New simulation approach

In this section, new ODE approximations are introduced for (3.1)(3.2) and (3.4)(3.5). Based

on those simple approximations, a general procedure is presented for approximating nons-

mooth mechanical systems involving many rigid-unilateral and Coulomb-frictional contacts.

3.2.1 Coulomb friction

The new approach for approximating (3.2) is motivated by Kikuuwe et al.’s work [69]. Their

work (specifically, model-C in [69]) provides an idea to approximate (3.2) by the following

DAI:

0 ∈ K(a + βa) − F sgn(p − a) (3.8a)

f = K(a + βa). (3.8b)


..

.Massless object

K¯

K

p¡af

p

p

p¡a a

stiffness

viscosity

Figure 3.3: A physical interpretation of (3.8).

Here, a ∈ Rr is a state variable newly introduced, K > 0 is a sufficiently large constant

and β > 0 is a constant appropriately chosen to suppress the oscillation in p. A physical

interpretation of the approximation (3.8) can be illustrated as Fig. 3.3. A friction force

described by F sgn(p−a) acts on a massless object whose velocity is p−a, and a viscoelastic

element with the stiffness K and the viscosity Kβ produces the force f in (3.8b), which

exactly balances the friction force.

In Kikuuwe et al.’s method, (3.8a) is discretized by Backward-Euler method, e.g., a is

replaced by (ak−ak−1)/T where T denotes the timestep size and the subscripts denote time

indices, and then it is analytically solved with respect to ak by using Theorem 1. In Bastien

and Lamarque’s model [71], a set of inclusions and equations with similar form to (3.8), are

also discretized by Back-Euler method and then analytically solved.

The observation that motivated the new approach is that, (3.8) can be solved without

using the Backward-Euler method. By the direct application of Theorem 1, (3.8) can be

equivalently rewritten as the following ODE:

a = (sat(F/K, a + βp) − a)/β (3.9a)

f = Ksat(F/K, a + βp). (3.9b)

As far as the author are aware, the literature includes no computational methods making

use of the equivalence between DAIs of the form of (3.8) and ODEs of the form of (3.9).


0 2 4 6 80.00

0.01

0.02

0.03

0.04

0 2 4 6 80.0

0.1

0.2

0.3

0.4

0.5

fe (

N)

(a)

time t (s)

p (

m)

(b)

time t (s)

Figure 3.4: Simulation of the system (3.1); (a) provided external force fe described as (3.11);

(b) simulation results by RK4 with the timestep size 0.001s. The parameters are chosen as;

M = 1 kg, F = 0.5 N, K = 5 × 103 N/m, β = 2 × 10−3 s. The initial conditions are;

p = 0 m, p = 0 m/s.

After replacing (3.2) by (3.9b) and appending (3.9a) to the state-space model, the system


(3.1)(3.2) is approximated by the following ODE:

d

dt

p

p

a

=

(fe − Ksat(F/K, a + βp))/M

p

(sat(F/K, a + βp) − a)/β

. (3.10)

Fig. 3.4 shows the simulation result by using the ODE (3.10) with RK4. To illustrate the

advantage of this method, it also presents the result of LuGre model (3.3) combined with

(3.1). In the simulation, an external force fe was chosen as

fe =

min(0.55, 0.5t) N if t < 2 s

0.35 + 0.15cos(100t) N otherwise,(3.11)

which is, after t = 2s, oscillatory below the static friction level F = 0.5 N. As shown in

Fig. 3.4(b), LuGre model produces unrealistic positional drift, which has been known in the

literature (e.g.,[69, 70]), while the presented method (3.10) does not. This implies that (3.9)

is a better approximation of (3.2) than (3.3).

It should be mentioned that, (3.9) is derived by relaxing the rigid constraint between

f and p in (3.2) by introducing an auxiliary variable a that has its own dynamics. In this

sense, the proposed method may be viewed to be similar to Baumgarte’s method [84], in

which constraints are relaxed to improve the numerical stability of the solutions of ODEs.

One of concerns regarding models based on DIs is the existence and uniqueness of solution,

as discussed by Bastien and Lamarque [71]. As for the case of (3.8), on the other hand, it is

clear because (3.8) is equivalent to the ODE (3.9).

3.2.2 Rigid unilateral contact

The new approach for approximating (3.5) is a modification of the work by Kikuuwe and

Fujimoto [80]. In their approach, (3.5) is approximated by the following DAI:

0 ∈ K(e + βe) − dio(p + e) (3.12a)


KMassless object

fp

p+e

K¯e

pp+e

.. .

Figure 3.5: A physical interpretation of (3.12)

f = K(e + βe) (3.12b)

where K and β are appropriate positive constants, e ∈ R is a state variable newly introduced.

A physical interpretation of (3.12) is illustrated as Fig. 3.5. Here, a massless object whose

position is p+e is connected to the mass through a viscoelastic element with the stiffness K

and the viscosity Kβ. Due to the contact, the contact force dio(p + e) acts on the massless

object and it balances the force f from the viscoelastic element. In Kikuuwe and Fujimoto’s

work, (3.12) was discretized by Backward-Euler method and then analytically solved by the

application of Theorem 2. Unfortunately, (3.12) cannot be rewritten into an ODE because e

cannot be obtained explicitly.

The new approach presented here is to add another term αe to the argument of dio(·),

which yields the following DAI:

0 ∈ K(e + βe) − dio(p + e + αe) (3.13a)

f = K(e + βe) (3.13b)

where α > 0 is another appropriate constant. By using Theorem 2, (3.13) can be equiva-

lently rewritten as the following ODE:

e = max(−e/β,−(p + e)/α) (3.14a)

f = Kmax(0, e − β(p + e)/α). (3.14b)


0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

time t (s)

p (

m)

Figure 3.6: Simulation of the system (3.16) by RK4 with the timestep size 0.001s. The

parameters are chosen as; M = 1 kg, fe = −9.8 N, K = 105 N/m, β = 0.01 s, α =0.01 s(gray dashed), 0.007 s(black dashed), 0.005 s(gray solid), 0.001 s(black solid). The

initial conditions are; p = 1 m, p = 0 m/s.

The equivalence between DAIs of the form of (3.13) and ODEs of the form of (3.14)

has not been pointed out in the literature either. One can see that (3.14) is continuous with

respect to p, p and e and it does not produce sucking force because the right-hand side

of (3.14b) is always positive. This features in contrast to the conventional methods (3.6)

and (3.7), which are discontinuous, and to Hunt and Crosley’s model [12, 55, 79], which

produces a sucking force. It is also easy to see that (3.13), or equivalently, of (3.14) has a

unique solution.

One possible interpretation of (3.13) and its equivalent expression (3.14) can be ex-

plained by defining e∆= e + αe. By using e, (3.13) can be rewritten as follows:

0 ∈ L−1

[

L[K(e + β ˙e)]

1 + αs

]

− dio(p + e) (3.15a)

f = L−1

[

L[K(e + β ˙e)]

1 + αs

]

. (3.15b)

where L denotes the Laplace transform. By noting the similarity between (3.15) and (3.12),


one can see that force f in (3.15b) can be interpreted as a low-pass filtered viscoelastic force

although it does not exist in the real world. When α = β, (3.15) is equivalent to (3.12)

with β = 0, which produces a perfectly elastic force. To preserve the effect of the viscous

force and stability, it is presumable that α should be set smaller than β, although any tuning

guidelines are not obtained yet.

By replacing (3.5) by (3.14b) and appending (3.14a) to the state-space model, the system

(3.4)(3.5) is approximated by the following ODE:

d

dt

p

p

e

=

(fe + Kmax(0, e − β(p + e)/α))/M

p

max(−e/β,−(p + e)/α)

. (3.16)

A set of numerical simulation of the ODE (3.16) was performed with different α val-

ues and a fixed β value. Fig. 3.6 shows that the bouncing motion becomes smaller as α

decreases. This is consistent with the interpretation based on (3.15), which implies that a

smaller α strengthens the viscous effect in a high-frequency region. Detailed analysis on

the relation between the parameter values and the achieved coefficient of restitution is left

outside the scope of this chapter. What can be said is that the coefficient of restitution can

be adjusted by appropriate choices of α and β on a trial-and-error basis.

3.2.3 Coulomb-frictional, rigid unilateral contact

The methods in Sec. 3.2.1 and Sec. 3.2.2 can be easily combined to describe a rigid unilateral

contact involving Coulomb friction. Let us consider a rigid mass M of which the position

is p ∈ R3 and a rigid frictional surface perpendicular to the z axis and including the origin.

Then, the state-space model of the system can be described as the following DI:

d

dt

p

p

∈

p

1

M

µdio(pz)sgn(pxy)

dio(pz)

(3.17)


where p =[

pTxy, pz

]T

, pxy ∈ R2 and µ is the friction coefficient between the mass and the

surface.

It must be noticed that (3.17) includes a multiplication of dio(·) and sgn(·). To integrate

it, one must replace dio(·) first and then replace sgn(·) because the replacement of sgn(·)

involves its multiplicative factor (F in (3.2)) while that of dio(·) can be done independently.

In conclusion, the DI (3.17) can be approximated by the following ODE:

d

dt

p

p

e

a

=

p

1

M

fxy(pz, pxy, e, a)

fz(pz, e)

max(−e/β1,−(pz + e)/α)

(fxy(pz, pxy, e, a)/K2 − a)/β2

(3.18)

where

fz(pz, e)∆= K1 max(0, e − β1(pz + e)/α) (3.19)

fxy(pz, pxy, e, a)∆= K2sat(µfz(pz, e)/K2β2pxy + a) (3.20)

and the parameters α, β1, β2, K1 and K2 are appropriate positive constants. This ODE is

obtained by replacing dio(pz) in (3.17) by fz(pz, e) and then replacing fz(pz, e)sgn(pxy) by

fxy(pz, pxy, e, a).

3.2.4 General procedure

Now we are in position to present the main contribution of the work. A mechanical system

can be generally described by a DI in the following form:

x ∈ Φ(x) (3.21)


where x ∈ Rn is the state vector of the system. Here, Φ is a function that contains dio(·)

and sgn(·) in several places and may also contain single valued functions. Let us assume

that, in Φ, m different arguments, denoted as ψi(x), i ∈ 1, · · · ,m, are used for dio(·)

and that l different arguments, denoted as θi(x), i ∈ 1, · · · , l, are used for sgn(·). Here,

ψi : Rn → R+ and θi : R

n → Rr, r ∈ 1, 2, are continuous functions.

By applying the methods introduced in Sec. 3.2.1 and 3.2.2, Φ(x) can be approximated

by the following procedure:

1. First, replace dio(ψi(x)) by Kdi max(0, ei − βdi(ψi(x) + ei)/αi), where Kdi, βdi and

αi are appropriate positive constants.

2. Next, let χi(x) denote the multiplicative factors of sgn(θi(x)), which are nonnegative

continuous functions. Then, replace χi(x)sgn(θi) by Ksisat(χi(x)/Ksi, βsiθi(x)+ai),

where Ksi and βsi are appropriate positive constants.

3. Finally, append ei = max(−ei/βdi,−(ψi(x) + ei)/αi), i ∈ 1, · · · ,m, and ai =

(sat(χi(x)/Ksi, ai + βsiθi(x)) − ai)/βsi, i ∈ 1, · · · , l, to the state-space model.

With this procedure, the nonsmooth system (3.21) is approximated by the following

ODE:

d

dt

x

e1

...

em

a1

...

al

=

Φ(x, e1, · · · , em, a1, · · · , al)

max(−e1/βd1,−(ψ1(x) + e1)/α1)

...

max(−em/βdm,−(ψm(x) + em)/αm)

(sat(χ1(x)/Ks1, a1 + βs1θ1(x)) − a1)/βs1

...

(sat(χl(x)/Ksl, al + βslθl(x)) − al)/βsl

(3.22)

where Φ(x, e1, · · · , em, a1, · · · , al) is the function Φ(x) in which the aforementioned re-

placements are made.


z

x

y

0

R

M

Figure 3.7: Example I: A rolling sphere with collision and slip. In the simulation, the

parameters are chosen as; M = 1 kg, R = 0.5 m, µ = 0.1 and the initial conditions are;

p = [5.5, 0, 0]T m/s, p = [0, 0, 2R]T m, ω = [0, 0, 0]T rad/s.

The presented procedure cannot apply if the function Φ(x) includes a sgn(·) whose

multiplicative factor involves discontinuous functions other than dio(·) and if χi(x) are not

guaranteed to be nonnegative. The author, however, are not aware of nonsmooth mechanical

systems that must be described by such Φ(x) functions. As illustrated in Sec. 3.3.2, the

presented procedure can deal with such a complicated system as the one in Fig. 3.9.

3.3 Examples and simulation results

3.3.1 Example I: A rolling sphere with collision and slip

The presented approach is now illustrated by examples. Let us consider a system in which

a spherical object with a uniform mass density falls onto a fixed rigid surface. The surface


:pz (

m/s

)px an

d p

x¡!y R

(m

/s)

!y (

rad/s

)

:

px¡!y R:

(b)

time t (s)

(d)time t (s)

(c)time t (s)

(a)time t (s)

pz (

m)

px

::

0 1 2 3 4

0

1

2

3

4

5

6

0 1 2 3 4

¡3

¡2

¡1

0

1

2

3

(x-velocity of the

bottom point on the ball)

(e)

time t (s)

(f)time t (s)

pz (

m)

0 1 2 3 40.5

0.6

0.7

0.8

0.9

1.0

px¡!y R

(m

/s)

:

2.4 2.6 2.8 3.0 3.2 3.4 3.6¡0.02

¡0.01

0.00

0.01

0.02

0 1 2 3 4

0

2

4

6

8

3.25 3.30 3.35 3.40 3.45 3.50 3.55

0.4998

0.4999

0.5000

0.5001

Figure 3.8: Simulation results of Example I by using (3.27) integrated by RK4 with the

timestep size 0.001 s. The parameters are chosen as; K1 = K2 = 1 × 105 N/m, β1 = β2 =4 × 10−3 s, α = 2.8 × 10−3 s.

includes the origin and is perpendicular to the z axis. This example is also introduced in [72]

and a similar example is employed in [12]. Let p ∈ R3 be the position of the gravity center

of the object, q be the unit quaternion representing the attitude of the object, and ω ∈ R3 be


the angular velocity of the object. Let R and M be the radius and the mass of the object,

respectively, and µ be the friction coefficient between the object and the surface. Then, the

equations of motion of the object can be described as the following DI:

d

dt

p

p

ω

q

∈

1

M

−µdio(pz − R)sgn(v(p, ω))

dio(pz − R)

− g

p

5

2MR2

d ×

−µdio(pz − R)sgn(v(p, ω))

dio(pz − R)

Q(ω, q)

(3.23)

where

v(p, ω)∆= [px − ωyR, py + ωxR]T , (3.24)

Q : R3 × R

4 → R4 denotes an appropriate function that transforms ω into the quaternion

rate q, d∆= [0, 0,−R]T and g

∆= [0, 0, 9.8m/s2]T .

According to the procedure presented in Sec. 3.2.4, the DI (3.23) can be approximated

by an ODE in the following procedure. First, one should replace dio(pz − R) by

ψ(pz, e)∆= K1 max(0, e − β1(e + pz − R)/α) (3.25)

where e ∈ R and K1, β1 and α are appropriate positive constants. Then µdio(pz−R)sgn(v(p, ω))

becomes µψ(pz, e)sgn(v(p, ω)). Next, µψ(pz, e)sgn(v(p, ω)) should be replaced by

θ(pz, v(p, ω), e, a)∆= K2sat(µψ(pz, e)/K2, a + β2v(p, ω)) (3.26)

where a ∈ R2 and K2 and β2 are positive constants appropriately chosen. Finally, ODEs

defining the behaviors of e and a should be appended to (3.23). Then, (3.23) is approximated

by the following ODE:


d

dt

p

p

ω

q

e

a

=

1

M

−θ(pz, v(p, ω), e, a)

ψ(pz, e)

− g

p

5

2MR2

d ×

−θ(pz, v(p, ω), e, a)

ψ(pz, e)

Q(ω, q)

max(−e/β1,−(e + pz − R)/α)

(θ(pz, v(p, ω), e, a)/K2 − a)/β2

. (3.27)

Fig. 3.8 shows the result of the simulation by using (3.27) with RK4. Here, K1 and K2

are set as high as possible to achieve small penetrations during collisions, and β1, β2 and α

are chosen based on some trials and errors. Fig. 3.8(a) and Fig. 3.8(b) show the bouncing

motion in the z direction and Fig. 3.8(c) and Fig. 3.8(d) show the transition from pure trans-

lation (slipping in contact) to pure rolling. One can see that those behaviors are properly

simulated, which supports the validity of the approximation (3.18) of the original DI (3.17).

3.3.2 Example II: Multiple friction with multiple rigid unilateral con-

tacts

Next example is the application of the presented method to a system involving many fric-

tional contacts interacting to one another. Let us consider a planar system illustrated in

Fig. 3.9, which consists of a conveyor moving at a constant velocity u, a spring with the

stiffness Ks, two rigid objects M1 and M2 and a rigid vertical wall. The object M1 can

move freely in the horizontal direction and is subject to the elastic force from a spring

Ks in the vertical direction. It is assumed that the objects do not rotate. The coefficients

of friction between the wall and M1, between M1 and M2 and between M2 and the con-

veyor are µ1, µ2 and µ3, respectively. The state vector is defined as x∆= [pT , pT ]T where


u

(p2x, p2y)

Ks

(p1x, p1y)M2

M1

Figure 3.9: Example II: Multiple friction contacts with multiple rigid unilateral contacts. In

the simulation, the parameters were chosen as; µ1 = µ2 = µ3 = 0.5, M1 = 0.5 kg, M2 =1 kg, Ks = 100 N/m, u = 1 m/s and the initial conditions are; p = [0, 0.25, 0.05, 0]T m,

p = [0, 0, 0, 0]T m/s.

p = [p1x, p1y, p2x, p2y]T where [p1x, p1y]

T and [p2x, p2y]T denote the positions of M1 and M2,

respectively. The object M1 is regarded to be at [0, 0]T when it is in contact with the wall

and the spring balances the gravity. The object M2 is regarded to be at [0, 0]T when it is in

contact with the conveyor and the object M2 being at its [0, 0]T . Then, the state-space model

of the system can be described as the following DI:

d

dt

p

p

∈

p

(dio(p1x) − dio(p2x − p1x))/M1

(−Ω1(x) − Ω2(x) − Ks1p1y)/M1

(−Ω3(x, u) + dio(p2x − p1x))/M2

(Ω2(x) + dio(p2y))/M2 − g

(3.28)


time t (s)

(b)

p1x a

nd

p2x

(m

)

(a)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.00

0.01

0.02

0.03

0.04

0.05p2x

p1x

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4¡0.8

¡0.6

¡0.4

¡0.2

0.0

0.2

0.4

0.6

p1x a

nd

p2x

(m

/s)

time t (s)

p2x

:

p1x

::

:

(c)

time t (s)

time t (s)p2y

(10¡

3m

) p1y (

m)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

¡0.10

¡0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4¡0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

(d)

Figure 3.10: Simulation results of Example II by using (3.35) integrated by RK4 with

timestep size 0.001s. The gray regions indicate the time periods in which the objects M1

and M2 are in contact with each other. The parameters are chosen as; Ki = 5 × 105 N/m,

βi = 2 × 10−3 s (∀i ∈ 1, · · · , 6), αi = 1.6 × 10−3 s (∀i ∈ 1, 2, 3).

where Ω1(x)∆= µ1dio(p1x)sgn(p1y), Ω2(x)

∆= µ2dio(p2x − p1x)sgn(p1y − p2y) and Ω3(x, u)

∆= µ3dio(p2y)sgn(p2x + u).

According to the procedure presented in Sec. 3.2.4, the DI (3.28) is approximated by

an ODE in the following procedure. First, one should replace dio(p1x), dio(p2x − p1x) and

dio(p2y) by

ψ1(x, e1)∆= K1max(0, e1 − β1(p1x + e1)/α1), (3.29)

ψ2(x, e2)∆= K2max(0, e2 − β2(p2x − p1x + e2)/α2), (3.30)


ψ3(x, e3)∆= K3max(0, e3 − β3(p2y + e3)/α3), (3.31)

respectively, where Ki, βi and αi (i ∈ 1, 2, 3) are appropriate positive constants. Then,

Ω1(x), Ω2(x) and Ω3(x, u) are found to be replaced by µ1ψ1(x, e1)sgn(p1y), µ2ψ2(x, e2)sgn(p1y−

p2y) and µ3ψ3(x, e3)sgn(p2x+u), respectively. Next, they should be replaced by θ1(x, e1, a1),

θ2(x, e2, a2) and θ3(x, u, e3, a3), respectively, where

θ1(x, e1, a1)∆=K4sat(µ1ψ1(x, e1)/K4, a1 + β4p1y), (3.32)

θ2(x, e2, a2)∆=K5sat(µ2ψ2(x, e2)/K5, a2 + β5(p1y − p2y)), (3.33)

θ3(x, u, e3, a3)∆=K6sat(µ3ψ3(x, e3)/K6, a3 + β6(p2x + u)), (3.34)

and Ki and βi (i ∈ 4, 5, 6) are appropriate constants. Finally, the differential equations

defining the behaviors of ei and ai (i ∈ 1, 2, 3) should be appended to (3.28). Then, (3.28)

is approximated by the following ODE:

d

dt

p

p

e1

e2

e3

a1

a2

a3

=

p

(ψ1(x, e1) − ψ2(x, e2))/M1

(−Ksp1y − θ1(x, e1, a1) − θ2(x, e2, a2))/M1

(−θ3(x, u, e3, a3) + ψ2(x, e2))/M2

(θ2(x, e2, a2) + ψ3(x, e3))/M2 − g

max(−e1/β1,−(p1x + e1)/α1)

max(−e2/β2,−(p2x − p1x + e2)/α2)

max(−e3/β3,−(p2y + e3)/α3)

(θ1(x, e1, a1)/K4 − a1)/β4

(θ2(x, e2, a2)/K5 − a2)/β5

(θ3(x, e3, a3)/K6 − a3)/β6

. (3.35)

A numerical simulation was performed by using the ODE (3.35) with RK4. The results


u

p

Fs (p)

M

Fc (p¡u)²

Figure 3.11: Example III: Periodic motion. In the simulation, the parameters were chosen

as; F = 0.2 N, M = 1 kg, k1 = 1 N/m, k2 = 1 N/m3, u = 1 m/s, V = 1 m/s, and the initial

conditions are adopted from [1] for comparison; p = 1.19149 m, w = 0 m/s.

are shown in Fig. 3.10. Here, again, Ki are set as high as possible to achieve small pen-

etrations during collisions, and βi and αi are chosen based on some trials and errors. The

time periods indicated by the gray regions are those in which the objects M1 and M2 are

in contact to each other. Fig. 3.10(a) shows the horizontal bouncing motion of M1 and M2,

which eventually converges. Fig. 3.10(b) shows the vertical motion of M1, which exhibits

nonsmooth changes in the velocity during the contact with M2, being influenced by the

friction force. The vertical position of M1 does not converge to zero because of the static

friction forces from the wall and M2. Fig. 3.10(c) shows the vertical motion of M2, which

determines the normal force from the conveyor to M2. It properly shows the influence in

the normal force from the friction force acting on the side face. Those results are physically

plausible and indicate the validity of the approximation (3.35) of the original DI (3.28).

3.3.3 Example III: Periodic motion

This section shows the application of the proposed method to a system exhibiting periodic

motion. Let us consider the system illustrated by Fig. 3.11, which has been investigated by

Awrejcewicz et al. [1]. Fig. 3.11 shows that a mass M , of which the position is denoted as

p ∈ R, rests on a conveyor rolling with a constant velocity u ∈ R. The mass is subjected


0 5 10 15 20 25

¡1

0

1

2

3

simple smoothing (3.41)

Time (s)

p (m)

w (

m/s

)p

(m

), w

(m

/s)

(a)

(b)

¡1.5 ¡1.0 ¡0.5 0.0 0.5 1.0 1.5¡0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

proposed (3.40)w

p

simple smoothing (3.41)

proposed (3.40)

Figure 3.12: Simulation results of Example III by using (3.40) and (3.41) integrated by RK4

with timestep size 0.001 s. The parameters in (3.40) are chosen as; K = 1 × 105 N/m,

β = 0.5 s. The parameter in (3.41) is chosen as ϵ = 10−5 m/s.

to a nonlinear spring force Fs(p) and a rate-dependent friction force Fc(p − u). Then, the

system is described as the following equation:

Mp + Fs(p) − Fc(p − u) = 0. (3.36)


Here, let us assume that Fs(p) and Fc(p − u) are defined as follows:

Fs(p)∆= −k1p + k2p

3 (3.37)

Fc(p − u)∆= −

F

V + |p − u|sgn(p − u) (3.38)

where k1, k2, F and V are positive constants. By using a new variable w∆= u − p, one can

rewrite (3.36) into the following DI:

d

dt

p

w

∈

u − w

1

M

(

k2p3 − k1p −

F

V + |w|sgn(w)

)

. (3.39)

According to the procedure presented in Section 3.2.4, (3.39) is approximated as follows:

d

dt

p

w

a

=

u − w

1

M

(

k2p3 − k1p − sat

(

F

V + |w|, Ka + Kβw

))

(

sat

(

F

(V + |w|)K, a + βw

)

− a

)

/β

(3.40)

where a is a new state variable and K and β are positive parameters. In contrast, Awre-

jcewicz et al. [1] used the following equation to approximate (3.39):

d

dt

p

w

=

u − w

1

M

(

k2p3 − k1p −

F sat(ϵ, w)

(V + max(|w|, ϵ))ϵ

)

(3.41)

where 0 < ϵ ≪ V is a parameter. This approximation was obtained by simply replacing the

discontinuity by a linear function of a constant slope in the region |w| < ϵ. Awrejcewicz et

al. [1] has shown that this approximation does reproduce periodic motion appropriately.

Fig. 3.12 shows the simulation results of the proposed approximation (3.40) and the

simple smoothing (3.41). It shows that the proposed approximation (3.40) also provides


periodic solution, and it is very close to that of the simple smoothing (3.41). Considering

that the result of (3.41) has been analytically validated through Tichonov theorem in [1],

one can see that the result of the new approximation (3.40) is also valid.

3.4 Summary

This chapter has introduced a new method to approximate DIs describing nonsmooth me-

chanical systems involving Coulomb friction and rigid unilateral contact by ODEs. A main

difference of the new method from conventional regularization methods is that the resultant

ODEs are equivalent to DAIs that are approximations of DIs. As a consequence, the ap-

proximated ODEs preserve important features of the original DIs such as static friction and

always-repulsive contact force. An algebraic procedure for yielding the ODE approxima-

tions has been presented and has been illustrated by using some examples.

Future research should address the theoretical and numerical studies on the influence of

the chosen parameters (K, α, β) on the system behavior. Currently there are no guidelines

for the choice of the parameter values, thus they have been chosen through trial and error in

the presented examples. In particular, the choice of α and β strongly influences the realized

coefficient of restitution. Theoretical or empirical relations between the parameter values

and the coefficient of restitution must be sought in the future study.

One limitation of the presented approach is that it is only for “lumped” contacts. In some

situations, the contact force may be distributed across a contact area. It is unclear whether

the presented approach is applicable or not to such situations. Anisotropic friction force and

elastic contact, such as those seen in vehicle tires, would demand further extension of the

presented approach.

Chapter 4

A friction model with realistic presliding

behavior

This chapter proposes a multistate friction model by extending the previous single-state

friction model introduced in Chapter 3. This new model is described as a set of continuous

differential equations that describe both the presliding and sliding regimes in a unified ex-

pression. It reproduces major features of friction phenomena reported in the literature, such

as the Stribeck effect, nondrifting property, stick-slip oscillation, presliding hysteresis with

nonlocal memory, and frictional lag. Moreover, the new model does not produce unbounded

positional drift or non-smooth forces, which are major problems of previous models due to

the mathematical difficulty in dealing with transitions between the presliding and sliding

regimes. The model is validated through comparison between its simulation results and

empirical results in the literature.

The rest of this chapter is organized as follows. Section 4.1 overviews some related mod-

els of friction. Section 4.2 proposes the multistate model. Section 4.3 provides theoretical

analysis on properties of the proposed model. Section 4.4 validates this model through sim-

ulations in comparisons with the GMS model in the literature. Finally, concluding remarks

⋆ Portions of the materials in this chapter are reprinted from Tribology Letters, vol. 51, no. 3, Xiaogang

Xiong and Ryo Kikuuwe and Motoji Yamamoto, “A Multistate Friction Model Described by Continuous

Differential Equations”, pp. 513–523, 2013, as published in [85, 86], with permission from Springer (see

Appendix A1) .

41

Chapter 4. A friction model with realistic presliding behavior 42

are given in Section 4.5.

4.1 Related work

4.1.1 LuGre model, single-state elastoplastic model, Leuven model and

modified Leuven model

The LuGre model without the Stribeck effect has been reviewed in Chapter 3.1.1 as an

approximation of the set-valued law (3.2). For easinesses of comparisons with other friction

models, this model is rewritten here with the Stribeck effect and new parameter notations

[21, 31]:

a = v

(

1 − sgn(v)κa

g(v)

)

(4.1a)

f = κa + σa. (4.1b)

Again, f is the friction force and v is the relative velocity between two surfaces in contact.

From the physical point of view, a should be interpreted as the average deflection of as-

perities and κ, σ ∈ R+ are the stiffness and micro-damping coefficients, respectively. For

simplicity and without loss of generality, the viscous component Dv of the friction force

f in (4.1b) is not considered throughout this chapter. Function g(v) describes the Stribeck

effect, which can be typically described as follows [21, 31]:

g(v)∆= Fc + (Fs − Fc)e

−|v/vs|α

. (4.2)

Here, Fc and Fs are the magnitude of kinetic friction force and static friction force, respec-

tively. The parameter α ∈ R+ and the Stribeck velocity vs are constants. The LuGre model

produces unbounded positional drift due to its inappropriate way of modeling the elastic

deformation in the presliding regime [32].

The single state elastoplastic model [32] does not suffer from the unbounded positional

drift problem. However, as pointed out in [35], this model produces unrealistic presliding

hysteresis behaviors that do not show nonlocal memory. The Leuven model [24, 34] pro-


......

v

·1¾1

·2¾2

·N

¾N

aN

a2

a1

Figure 4.1: A physical interpretation of (4.3b) and (4.3b). In the sliding regime, the asperity

deflections ai evolve so that the elastic forces |κiai| converge to λig(v). The total friction

force f is the sum of the viscoelastic forces of the elements. Figure and caption are reprinted,

with permission from Springer: Tribology letter, “A Multistate Friction Model Described

by Continuous Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong,

Ryo Kikuuwe, and Motoji Yamamoto, Fig. 2 (see Appendix A1).

duces presliding hysteresis behaviors with nonlocal memory in the presliding regime, but

as pointed out in [35], it has implementation difficulties since it requires additional stacks

to store extreme force values. In contrast, the modified Leuven model [34] is free from

those implementation difficulties. However, as pointed out and illustrated in [23, 35, 40],

the force-position and force-velocity curves produced by this model do not exhibit an ap-

propriate nondrifting property and frictional lag effect that agree with experimental data.

4.1.2 Generalized Maxwell-Slip friction model

The generalized Maxwell-slip (GMS) model [35] is a friction model that can be represented

by a parallel connection of N elastoplastic elements, as illustrated in Fig. 4.1. This model is


described as follows [35–37]:

f =N

∑

i=1

(κiai + σiai), i ∈ 1, · · · , N (4.3a)

ai =

v if si = STICK

Cλisgn(v)

κi

(

1 − sgn(v)κiai

λig(v)

)

if si = SLIP(4.3b)

where si are binary state variables, each of which remains STICK until |ai| = λig(v)/κi and

remains SLIP until v goes through zero. Here,∑N

i=1 λi = 1 and κi, σi ∈ R+ are the stiffness

and micro-damping coefficients of the ith element, respectively. The state variable ai of the

ith element can be interpreted as the deflection of an asperity as is the case with the LuGre

model. The positive constant parameter C, which replaces a quantity proportional to |v| in

the LuGre model (4.1a)(4.1b) determines how fast ai converges to λig(v)/κi.

The GMS model is capable of capturing many features of friction phenomena, such as

presliding hysteresis with nonlocal memory, frictional lag, stick-slip oscillation, and non-

drifting property [23, 35, 40]. One of its drawback is that the switching structure, involving

additional binary variables si, results in the difficulty in on-line parameter identification, as

pointed out in [36, 37]. Another point that should be noted is that, when σi > 0, the force

f produced by (9) can be discontinuous with respect to time since (9b) does not guarantee

the continuity of a with respect to time. Such a discontinuity, however, has not been seen

as a problem. In fact, many previous works [87–90] employ a version of the GMS model

without the term σiai. In [36, 37], a smoothed version of the GMS model is proposed, but

it involves another two functions and three parameters, which further increase the difficulty

in parameter identification.

4.1.3 Differential-algebraic single-state friction model

In Chapter 3, the author proposed a single-state friction model based on a differential-

algebraic inclusion (DAI). This model is only valid for the case where the magnitudes of

maximum static friction force and kinetic friction force are equal to each other. It is derived


.v¡a

v

·

¾

asperity tip

Fc Sgn(v¡a)

.

a

Figure 4.2: A physical interpretation of the model (4.4a) and (4.4b). The viscoelastic force

balances the friction force FcSgn(v − a) both in the sliding and presliding regimes. Figure

and caption are reprinted, with permission from Springer: Tribology letter, “A Multistate

Friction Model Described by Continuous Differential Equations”, vol. 51, no. 3, pp. 513–

523, 2013, Xiaogang Xiong, Ryo Kikuuwe, and Motoji Yamamoto, Fig. 3 (see Appendix

A1).

from the following DAI:

0 ∈ κa + σa − FcSgn(v − a) (4.4a)

f = κa + σa. (4.4b)

Here, the friction force acting on the averaged asperity is determined by a set-valued Coulomb

friction law FcSgn(v − a) according to the velocity v − a of the asperity tip, of which the

deflection is a, as shown in Fig. 4.2. Equation (4.4a) means that the force balances (i.e., is

equal to) the force κa + σa produced by the viscoelasticity of the asperity.

It is clear that the DAI (4.4a)(4.4b) cannot be used for numerical computation of the

friction force f because it involves set-valued function Sgn(·). Our observation presented in

[33] is that, by the use of Theorem 1, the DAI (4.4a)(4.4b) is equivalently rewritten as the

following ordinary differential equation (ODE):

a =sat(Fc, κa + σv) − κa

σ(4.5a)

f = sat(Fc, κa + σv). (4.5b)

There are some friction models that have similar structures to (4.4a)(4.4b), which de-


scribe the Coulomb-like discontinuous relation between the force f and the asperity-tip

velocity v − a [69, 91, 92]. They however need to be numerically integrated through

through Euler methods. In contrast, because the model (4.4a)(4.4b) is equivalent to the

ODE (4.5a)(4.5b), it can be numerically integrated through any integration methods such as

Runge-Kutta methods.

In contrast to the Dahl and LuGre models, which are also described by ODEs, the model

(4.5a)(4.5b) (or equivalently, (4.4a)(4.4b)) does not produce positional drift, as demon-

strated in [33]. Obvious limitations of (4.5a)(4.5b) are that this model does not include

the Stribeck effect and that it cannot produce presliding hysteresis with nonlocal memory.

Those limitations will be treated in the next section.

4.2 New multistate friction model

4.2.1 Extension to include Stribeck effect

The previous friction model (4.4a)(4.4b), or equivalently, (4.5a)(4.5b), does not include

the Stribeck effect represented by the function g(v) in (4.2). One way to overcome this

limitation is to replace Fc by g(v), which represents rate-dependent friction force. Then,

one can extend (4.4a)(4.4b) into the following DAI:

0 ∈ κa + σa − g(v)Sgn(v − a) (4.6a)

f = κa + σa. (4.6b)

As was the case with (4.4a)(4.4b), by the direct application of Theorem 1, (4.6) can also

be equivalently rewritten into the following ODE:

a =sat(g(v), κa + σv) − κa

σ(4.7a)

f = sat(g(v), κa + σv). (4.7b)

Some differences and similarities among the model (4.7a)(4.7b), the LuGre model (4.1a)(4.1b),

and the GMS model (4.3b)(4.3b) are now discussed. Because the model (4.7a)(4.7b) does


not consider the frictional lag, it is in the presliding regime when the friction force κa + σv

satisfies |κa+σv| ≤ g(v). In this case, sat(g(v), κa+σv) in (4.7a)(4.7b) reduces to κa+σv

and thus (4.7a)(4.7b) reduces to the following equation:

a = v (4.8a)

f = κa + σv. (4.8b)

One can observe that (4.8) is equivalent to the GMS model (4.3b)(4.3b) in the presliding

regime with N = 1.

In the sliding regime, the steady-state friction force (i.e., the force f when v is con-

stant and a = 0) can be obtained by substituting a = 0 into (4.1a)(4.1b) (4.3b)(4.3b), and

(4.7a)(4.7b). In the case of LuGre model (4.1a)(4.1b) and GMS models (4.3b)(4.3b), one

can easily see that f = sgn(v)g(v) is satisfied in the steady state because both (4.1b) and

(4.3b) imply that a = 0 results in κa = sgn(v)g(v) 1. In the case of the model (4.7a)(4.7b),

|κa + σv| > g(v) (4.9a)

f = sgn(κa + σv)g(v) = κa (4.9b)

are satisfied in the steady state in the sliding regime. Eliminating g(v) yields |κa + σv| >

|κa|, which leads to sgn(a) = sgn(v) = sgn(κa + σv). Therefore, one can see that the

model (4.7a)(4.7b) also results in f = sgn(v)g(v) in the steady state.

The state variable a and its derivative a are both necessary to realize such a pair of in-

terchangeable expressions; the DAI (4.6a)(4.6b) and the ODE (4.7a)(4.7b). From a physical

point of view, a can be understood as the deflection of a linearly viscoelastic asperity.

4.2.2 Extension to include presliding hysteresis with nonlocal memory

The modified model (4.7a)(4.7b) represents a single viscoelasto-plastic element with rate-

dependent friction force, which is unable to produce hysteresis with nonlocal memory. As

has been suggested in the literature [34, 35, 43–45], one imaginable way to overcome this

1Here we are using N = 1 and λ1 = 1 in (4.3b)(4.3b) for the simplicity of comparison.


limitation is to extend (4.6a)(4.6b) (or equivalently, (4.7a)(4.7b)) to a multistate model com-

posed of many viscoelasto-plastic elements. Based on this idea, (4.6a)(4.6b) can be extended

as follows:

0 ∈ κiai + σiai − λig(v)Sgn(v − ai), i ∈ 1, · · · , N (4.10a)

f =N

∑

i=1

(κiai + σai). (4.10b)

A physical interpretation of the multistate DAI (4.10a)(4.10b) can be illustrated as Fig. 4.3,

which can be thought of as a straightforward extension of Fig. 4.2.

As was the case between (4.6a)(4.6b) and (4.7a)(4.7b), (4.10a) and (4.10b) are equivalent

to the following ODE:

ai =sat(λig(v), κiai + σiv) − κiai

σi

, i ∈ 1, · · · , N (4.11a)

f =N

∑

i=1

sat(λig(v), κiai + σiv). (4.11b)

This model produces presliding hysteresis with nonlocal memory.

4.2.3 Extension to include frictional lag

The multistate model (4.11a)(4.11b) does not capture frictional lag [25]. As has been sug-

gested in [12], the lag effect can be included by using an additional state variable as follows:

γ = (g(v) − γ)/τd (4.12a)

0 ∈ κiai + σiai − λiγSgn(v − ai), i ∈ 1, · · · , N (4.12b)

f =N

∑

i=1

(κiai + σai), (4.12c)

or equivalently,

γ = (g(v) − γ)/τd (4.13a)


....

..

v

·1

¾1

·2

¾2

·N

¾N

aN

a2

a1

¸Ng(v)Sgn(v¡aN).

¸2g(v)Sgn(v¡a2).

v¡aN.

v¡a2.

v¡a1.¸1g(v)Sgn(v¡a1)

.

Figure 4.3: A physical interpretation of the model (4.10a) and (4.10b). Figure and cap-

tion are reprinted, with permission from Springer: Tribology letter, “A Multistate Friction

Model Described by Continuous Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013,

Xiaogang Xiong, Ryo Kikuuwe, and Motoji Yamamoto, Fig. 4 (see Appendix A1).

ai =sat(λiγ, κiai + σiv) − κiai

σi

, i ∈ 1, · · · , N (4.13b)

f =N

∑

i=1

sat(λiγ, κiai + σiv). (4.13c)

Here, γ ∈ R is a new state variable representing a lagged value of g(v). Its initial value

should be set within the range Fc ≤ γ(0) ≤ Fs. The new parameter τd > 0 in (4.12a)

represents the time constant of convergence of γ to g(v). When τd approaches zero, the

model (4.12a)(4.12b)(4.12c) becomes close to (4.10a)(4.10b).

The proposed model (4.12a)(4.12b) (4.12c) is physically straightforward, only describ-


ing the balance between the linear viscoelastic force κa+σa and the friction force γSgn(v−

a), which depends on the first-order lagged value γ of g(v). It however has not been found

in the literature. One difficulty may have been that it is numerically intractable due to the

inclusion of the discontinuous Sgn(·) function. The difficulty is now removed by the use

of Theorem 1, which implies that the DAI (4.12a)(4.12b)(4.12c) is equivalent to the ODE

(4.13a)(4.13b)(4.13c) having no numerical difficulty.

4.3 Theoretical analysis

This section shows that the proposed model (4.13a)(4.13b)(4.13c) has the following proper-

ties, which should be possessed by a well-behaved friction model [35]:

1. The non-viscous part of the friction force is bounded.

2. The friction force is continuous with respect to time.

3. The friction model is dissipative.

To prove the first property of the proposed model (4.13a)(4.13b)(4.13c), a nonnegative

function is defined as V1∆= γ2/2. The time derivative of this function is as follows:

V1 = γγ = γ(g(v) − γ)/τd, (4.14)

which is negative when |γ| ≥ Fs due to Fc ≤ g(v) ≤ Fs. Therefore, one can obtain that

|γ| ≤ Fs is always satisfied if the initial value of γ satisfies |γ| ≤ Fs [21, 31]. From (4.13c),

one can see that |f | ≤∑N

i=1 λi|γ| ≤ Fs∑N

i=1 λi = Fs. This means that |f | is bounded.

The second property of the proposed model (4.13a)(4.13b) (4.13c) is obvious since the

function sat is continuous with respect to its arguments and the variables γ, a, and v are

continuous with respect to time.

The third property can be proven by showing that there exists a constant ϕ > 0 such

that∫ t0 vfdt ≥ −ϕ is satisfied for all t > 0 [21, 31]. Let us define a function V2 as V2

∆=


∑Ni=1 κia

2i /2. Then, one obtains

vf − V2 =N

∑

i=1

(σiv − κiai)sat(λiγ, κiai + σiv) + κ2i a

2i

σi

. (4.15)

When z = 0, the right hand side of (4.15) is positive. When z = 0, (4.15) reduces to the

following equation:

vf − V2 =N

∑

i=1

σ2i v

2 + κ2i a

2i (max(1, |κiai + σiv|/(λiγ)) − 1)

σi max(1, |κiai + σiv|/(λiγ))≥ 0. (4.16)

Therefore, one can see that vf − V2 ≥ 0 is always satisfied, and thus

∫ t

0

vfdt ≥ V2(t) − V2(0) > −V2(0) (4.17)

is also satisfied for all t > 0. This shows that the model (4.13a)(4.13b)(4.13c) has the third

property.

4.4 Simulation and comparison

This section illustrates the properties of the new model through simulations. Section 4.4.1,

4.4.2, and 4.4.3 provide the simulation results of the new model (4.13a)(4.13b)(4.13c). Sec-

tion 4.4.4 gives comparison between the GMS model and the new model. In all simulations,

the parameters of the models were chosen through trial-and-error so that they produce major

features of friction phenomena. It must be noted that an efficient parameter tuning method

remains an open problem. The forward Euler method was used for all simulations.

4.4.1 Nondrifting and stick-slip oscillation

It is well known that an external force that is smaller than the magnitude of static friction

force does not produce unbounded positional drift, which is termed as nondrifting property

[21, 22]. A simulation was performed to test this nondrifting property of the new model

(4.13a)(4.13b)(4.13c). In this simulation, an external force shown in Fig. 4.4(a) was applied


0 2 4 6 8 10

¡0.0

1

2

3

4

Exte

rnal

forc

e ((

N)

Time t (s)(a)

Position p (µm)

(b)

Fri

ctio

n f

orc

e f (

N)

0.0 0.2 0.4 0.6 0.8

¡1.0

¡0.5

0.0

0.5

1.0

1.5

2.0

2.5

Figure 4.4: Nondrifting property; (a) an external force as a function of time t; (b) the friction

force f as a function of position p. The parameters are chosen as; α = 2, vs = 5 µm/s, Fc =1 N, Fs = 2.5 N, κi = (5/i) N/µm, σi = 0.02 Ns/µm, λi = 0.1, i ∈ 1, 2, · · · , 10, τd =0.02 s. The time-step size was 0.001 s. Figure and caption are reprinted, with permission

from Springer: Tribology letter, “A Multistate Friction Model Described by Continuous

Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong, Ryo Kikuuwe,

and Motoji Yamamoto, Fig. 4 (see Appendix A1).


MvcK

vp

f

Figure 4.5: The scenario of stick-slip oscillation

to a mass M = 1 kg, which is subjected to friction force f produced by the new model

(4.13a)(4.13b)(4.13c). Fig. 4.4(b) shows the result, in which the position oscillates without

shift, exhibiting the nondrifting property of the new model.

Another set of simulation was performed to check the stick-slip behavior of the new

model. In this simulation, as shown in Fig. 4.5, the mass M = 1 kg, subjected to the friction

force f , was connected to a spring with the stiffness K = 0.1 N/µm, and the other end of

the spring was moved at a constant velocity vc = 2 µm/s. Fig. 4.6 shows the results, in

which one can observe that the mass M experienced a stick-slip motion. Fig. 4.6(a) shows

step-like changes in the position and impulsive changes in the velocity. The dashed circles

in Fig. 4.6(b) indicate spike-like force followed by high frequency oscillation after stick-slip

periods. Those phenomena are consistent with the empirical results [22]. They are known to

be reproducible with the GMS model (4.3b)(4.3b) and the generic models in [38, 39], while,

as pointed out in [35], the LuGre model [21, 31], the single state elastoplastic model [32],

and the Leuven model [24, 34] produce qualitatively different stick-slip motion.

4.4.2 Rate independency and amplitude dependency of hysteresis loop

Experimental data in the literature, e.g., [22, 39, 93], show that, in the presliding regime,

the hysteresis loop of a force-position curve is hardly affected by the change rate of po-

sition. This characteristic is referred to as rate independency of hysteresis loop. In con-

trast, different strokes of position input result in different loop sizes of force-position curves

[22, 39]. Here we refer to such a characteristic as an amplitude dependency of hystere-


0 50 100 1500

50

100

150

200

250

300

Vel

oci

ty v

(µ

m/s

)

0

5

10

15

0 50 100 150

¡2

¡1

0

1

2

3

4

Time t (s)

(b)

Time t (s)

(a)

Posi

tion p

(µ

m)

Forc

e (

N)

friction force spring force

20

25

30position velocity

Figure 4.6: Stick-slip oscillation; (a) the position and velocity of the mass M as a function

of time t; (b) the spring force and friction force as a function of time t. The parameters

were chosen identical to those in Fig. 4.4. The time-step size was 0.005 s. Figure and cap-

tion are reprinted, with permission from Springer: Tribology letter, “A Multistate Friction

Model Described by Continuous Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013,

Xiaogang Xiong, Ryo Kikuuwe, and Motoji Yamamoto, Fig. 5 (see Appendix A1).


0 20 40 60 80

¡1.0

¡0.5

0.0

0.5

1.0

1.0 0.5 0.0 0.5 1.0¡3

¡2

¡1

0

1

2

3

Posi

tion p (µ

m)

Time t (s)

(a)

Position p (µm)

(b)

Fri

ctio

n f

orc

e f (

N)

Figure 4.7: Rate independency of hysteresis loop; (a) two positional inputs p as functions

of time t (The gray curve is of 10 times the frequency of the black curve.); (b) the friction

force f obtained for the two different input position signals p. The parameters were chosen

identical to those in Fig. 4.4. The time-step size was 0.01 s. Figure and caption are reprinted,

with permission from Springer: Tribology letter, “A Multistate Friction Model Described

by Continuous Differential Equations”, vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong,



sis loop. Two simulations were performed to test these two properties of the new model

(4.13a)(4.13b)(4.13c).

In the simulation of rate independency of hysteresis loop, the positional inputs shown

in Fig. 4.7 (a) were provided to the new model and the resultant friction forces f were

recorded. The position inputs in Fig. 4.7(a) were chosen so that the presliding hysteresis with

nonlocal memory becomes visible. The gray curve in Fig. 4.7(a) is 10 times the frequency

of the black one. The result is shown in Fig. 4.7(b), which clearly shows that the model

(4.13a)(4.13b)(4.13c) is rate independent in the shape of the hysteresis loop. It also shows

that the force-position curves have closed internal loops whose top tips overlap the trajectory

of outer loop, implying that the new model captures the feature of presliding hysteresis with

nonlocal memory.

In the simulation of amplitude dependency of hysteresis loop, the positional inputs in

Fig. 4.8(a) were provided to the new model (4.13a)(4.13b)(4.13c), and the resultant forces

were plotted as a function of the positional inputs, as shown in Fig. 4.8(b). It is shown that

the sizes of the loops in the force-position hysteresis vary depending on different magnitudes

of input. This characteristic is consistent with the experimental results in [22, 39] and the

simulation results of the generic model presented in [39].

4.4.3 Frictional lag

It has been known that frictional lag results in hysteresis in the force-velocity curve, in

which the friction force during acceleration is larger than that during deceleration [23, 40].

A simulation was performed to check this property of the new model (4.13a)(4.13b)(4.13c).

In this simulation, two biased sinusoidal velocity inputs in Fig. 4.9(a) were provided to the

new model (4.13a)(4.13b)(4.13c). Fig. 4.9(b) shows the corresponding results, in which the

magnitude of friction force is larger during acceleration than during deceleration. Fig. 4.9(b)

also shows that the hysteresis loops of the force-velocity curves are influenced both by the

input frequencies and τd. The frequency effect on the hysteresis loops has been observed

in Hess and Soom’s experiments [94] and pointed out in Al-Bender et al.’s [35] and De

Moerlooze et al.’s modeling studies [39].

Another simulation was performed to investigate the effect of frictional lag on the new


model’s behavior during transitions between the presliding and sliding regimes. In this

simulation, two sinusoidal velocity inputs in Fig. 4.10(a) were provided to the new model

(4.13a)(4.13b)(4.13c). Fig. 4.10(b) shows the results. One can observe that the width of the

force-velocity loop increases as the input frequency increases. This feature is consistent with

experimental results in the literature [35, 40]. Fig. 4.10(b) also shows that, by appropriate

choice of τd, the new model can be adjusted so that the friction force during acceleration is

bigger than that in deceleration. This feature of the curves is reported in the experimental

results in the literature [22, 23, 40].

4.4.4 Comparison between GMS model and new model

The GMS model (4.3b)(4.3b) and the new model (4.13a)(4.13b)(4.13c)were compared through

a simulation of their transition behaviors. In this simulation, a sinusoidal velocity input

shown in Fig. 4.11(a) was provided to the two models. The parameters of the new model

were identical to those in Fig. 4.4. except three different values of σi (i = 1, · · · , 10) were

used in this simulation. The parameters of GMS model were also identical to those of the

new model except that the parameter C was chosen as C = 62 N/s. This C value was cho-

sen so that the resultant force-velocity curve becomes as close as possible to that of the new

model with σi = 0.001 Ns/µm, which are shown in Fig. 4.11(b).

The results are shown in Fig. 4.11(b) to (h). Fig. 4.11(b), (c), and (d) show that the force

produced by the GMS model is in some places nonsmooth, and this feature is magnified

as σi increases. Fig. 4.11(e) and (f) show that the discontinuities in the force coincide with

transitions between the presliding and sliding regimes of the elements. It is reasonable to

consider that this is caused by the equation (4.3b), which shows that ai is not guaranteed

to be continuous with respect to v and ai. The magnified discontinuities with larger σi are

also reasonable results because (4.3b) implies that the discontinuities in ai is propagated to

f through the coefficient σi. This sort of results have not been reported in previous result,

presumably because this effect is not apparent with sufficiently small σi values. Actually,

most of previous works [36, 37, 87–89] employing the GMS models ignore the term σiai

and thus the discontinuities have not been a concern. As for the proposed model, in contrast,

σi > 0 is necessary because (4.13a)(4.13b)(4.13c) includes the division by σi. Therefore,


one can say that the σi values should be sufficiently low in the GMS model but should be

sufficiently high in the proposed model.

As an effect of the continuous ODE (4.13a)(4.13b)(4.13c), the friction force produced

by the new model is always continuous with respect to the velocity and the time, as can be

seen in Fig. 4.11(b)(c)(d) and Fig. 4.11(g)(h), although some non-smoothness (discontinu-

ous changes in the rate-of-change of the force) can be observed in Fig. 4.11(g)(h). Another

feature of the new model can be seen in Fig. 4.11(g) and (h), in which the transition from

the sliding regime to the presliding regime occur at one element after another. This is in

contrast to the simultaneous transitions exhibited by the GMS model, shown in Fig. 4.11(e)

and (f).

4.5 Summary

The chapter has introduced a new multistate friction model that is an extended version of

the author’ differential-algebraic single-state friction model. This multistate friction model

consists of multiple parallel elements. Each element is described by a single-state friction

model, which exhibits smooth transitions between the presliding and sliding regimes. More-

over, the new model involves a parameter τd to adjust the effect of frictional lag, which is

the time lag from the change in the velocity to the change in the friction force. This model

is described as a set of continuous ODEs, and thus the friction force produced by this model

is always continuous with respect to time and the input velocity. It does capture many major

features of friction reported in the literature, such as the Stribeck effect, nondrifting prop-

erty, stick-slip oscillation, presliding hysteresis with nonlocal memory, rate independency

and magnitude dependency of hysteresis loop, and frictional lag. This model has been vali-

dated through simulations and comparisons with experimental results in the literature.

Potential applications of this model include not only simulation but also control for

friction compensation, because this model, i.e., the ODE (4.13a)(4.13b)(4.13c), is compu-

tationally inexpensive enough for realtime computation and does not require any iterative

computation. Future work should clarify a method to identify the parameters of the pro-

posed model. It may be possible to apply some on-line parameter identification methods


in the literature, e.g., [36, 37, 87], since this model is described by a set of continuous dif-

ferential equations (4.13a)(4.13b)(4.13c). Effects of each parameters on major features of

friction should also be clarified for efficient off-line tuning based on experimental data.

As another important topic of study, the proposed model should be experimentally com-

pared to previous friction models, such as the GMS model. One suitable approach may be,

as can be found in [95, 96], to optimize the parameters of each model to fit experimental data

from real friction phenomena, and to compare residual fitting errors obtained from the mod-

els. Relating such results to the complexity of the models, such as the number of parameters,

memory usage, and the computational cost, may lead to a set of guidelines for choosing a

friction model. One salient feature of the proposed model is that the micro-damping coeffi-

cient requires to be non-zero and can be rather high without affecting the continuity, while

it is usually considered zero in previous work. Practical advantages or disadvantages of this

feature may also be clarified by comparative study based on parameter fitting.


0 2 4 6 8¡1.0

¡0.5

0.0

0.5

1.0

1.0 0.5 0.0 0.5 1.0¡3

¡2

¡1

0

1

2

3

Fri

ctio

n f

orc

e f (

N)

Posi

tion p (µ

m)

Time t (s)

(a)

Position p (µm)

(b)

Figure 4.8: Amplitude dependency of hysteresis loop; (a) four positional inputs p with

different amplitudes as functions of time t; (b) the friction forces f obtained for the four

different input position signals p. The parameters were chosen identical to those in Fig. 4.4.

The time-step size was 0.001 s. Figure and caption are reprinted, with permission from

Springer: Tribology letter, “A Multistate Friction Model Described by Continuous Differ-

ential Equations”, vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong, Ryo Kikuuwe, and

Motoji Yamamoto, Fig. 7 (see Appendix A1).


0.0 0.5 1.0 1.5 2.0

0

5

10

15

20

25

0 5 10 15 20 25

1.0

1.5

2.0

2.5

Vel

oci

ty v (µ

m/s

)

Time t (s)

(a)

Velocity v (µm/s)

(b)

Fri

ctio

n f

orc

e f (

(N)

acceleration

deceleration

1Hz, ¿d=0.08s10Hz, ¿d=0.02s10Hz, ¿d=0.08s

g(v)

1Hz, ¿d =0.02s

Figure 4.9: Frictional lag; (a) two velocity inputs v as functions of time t; (b) the friction

force f obtained for the two different input velocity signals v and two values of τd. The other

parameters were chosen identical to those in Fig. 4.4. The time-step size was 0.001 s. Figure

and caption are reprinted, with permission from Springer: Tribology letter, “A Multistate

Friction Model Described by Continuous Differential Equations”, vol. 51, no. 3, pp. 513–

523, 2013, Xiaogang Xiong, Ryo Kikuuwe, and Motoji Yamamoto, Fig. 8 (see Appendix

A1).


0.0 0.5 1.0 1.5 2.0

¡10

¡5

0

5

10

¡10 ¡5 0 5 10

¡2

¡1

0

1

2

Vel

oci

ty v (µ

m/s

)F

rict

ion f

orc

e f (

(N)

Velocity v (µm/s)

(b)

Time t (s)

(a)

1Hz, ¿d =0.02s 1Hz, ¿d=0.08s

10Hz, ¿d=0.02s10Hz, ¿d=0.08s

Figure 4.10: Frictional lag effect on transition behavior; (a) two velocity input v as functions

of time t; (b) the friction force f obtained for the two different input velocity signals v and

two values of τd. The other parameters were chosen identical to those in Fig. 4.4. The time-

step was 0.001 s. Figure and caption are reprinted, with permission from Springer: Tribol-

ogy letter, “A Multistate Friction Model Described by Continuous Differential Equations”,

vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong, Ryo Kikuuwe, and Motoji Yamamoto,

Fig. 9 (see Appendix A1).


0.0 0.2 0.4 0.6 0.8 1.0 1.2

−10

−5

0

5

10

−10 −5 0 5 10

−2

−1

0

1

2

Friction force (N)

Friction force f (N)


Velocity (µm/s)

Time t (s)

(a)

Velocity v (µm/s)

(b)

Velocity v (µm/s)

(c)

Velocity v (µm/s)

(d)

GMS New

¾i=0.001 Ns/µm

(i21,2,¢¢¢,10)

GMS New

¾i=0.01 Ns/µm

(i21,2,¢¢¢,10)

GMS New

¾i=0.03 Ns/µm

(i21,2,¢¢¢,10)

−10 −5 0 5 10

−2

−1

0

1

2

10 5 0 5 10

−2

−1

0

1

2

Figure 4.11: Transition behaviors of the GMS model (4.3b)(4.3b) and new model (4.13a)

(4.13b) (4.13c); (a) a velocity input v as a functions of time t; (b)(c)(d) the friction force

f obtained from the input velocity v; (e)(f)(g)(h) the friction force f and the number of

elements in the presliding regimes; The gray vertical lines in (e)(f)(g)(h) indicate the time

at which the number of elements in the presliding regime changes. The parameters were

chosen identical to those in Fig. 4.4. The time-step size was 0.0001 s. (It should be noted

that non-zero σi has not been used in reported applications of the GMS model in the litera-

ture.) Figure and caption are reprinted, with permission from Springer: Tribology letter, “A

Multistate Friction Model Described by Continuous Differential Equations”, vol. 51, no. 3,

pp. 513–523, 2013, Xiaogang Xiong, Ryo Kikuuwe, and Motoji Yamamoto, Fig. 10 (see

Appendix A1).


0.20 0.25 0.30 0.35 0.40 0.20 0.25 0.30 0.35 0.40

−1

0

1

2

−2


−1

0

1

2

−2


0

5

10

No. of elem

s. in presliding

0

5

10

No. of elem

s. in presliding

0.20 0.25 0.30 0.35 0.40 0.20 0.25 0.30 0.35 0.40

−1

0

1

2

−2


−1

0

1

2

−2Friction force f (N)

0

5

10No. of elem

s. in presliding

0

5

10

No. of elem

s. in presliding

Time t (s)

(e)Time t (s)

(f)

Time t (s)

(g)

Time t (s)

(h)

force

¾i=0.01 Ns/µm

force

¾i=0.03 Ns/µm

force

¾i=0.01 Ns/µm

force

¾i=0.03 Ns/µm

Number

Number

Number

Number

GMS model GMS model

New model New model

Figure 4.11: (Continued.) Figure and caption are reprinted, with permission from Springer:

Tribology letter, “A Multistate Friction Model Described by Continuous Differential Equa-

tions”, vol. 51, no. 3, pp. 513–523, 2013, Xiaogang Xiong, Ryo Kikuuwe, and Motoji Ya-

mamoto, Fig. 10 (see Appendix A1).

Chapter 5

A contact model with nonlinear

compliance

This chapter proposes a compliant contact model with nonzero indentation based on the pre-

vious simple contact model in Chapter 3. The contact force and indentation of this model

possesses the following three features: (i) continuity of the force at the time of collision,

(ii) Hertz-like nonlinear force-indentation curve, and (iii) non-zero indentation at the time

of loss of contact force. On the contrary, the conventional Hunt-Crossley (HC) model does

not capture the feature (iii) as the model makes the contact force and the indentation reach

zero simultaneously. The comparisons between the HC model and the new model are illus-

trated through their force-indentation curves and an example of a free-falling ball. The be-

haviors of the new model and the effect of parameters in the model are investigated through

numerical simulations.

The rest of this chapter is organized as follow. Section 5.1 introduces some mathematical

preliminaries to be used in the subsequent sections. Next, some related work is outlined

in Section 5.1. Section 5.2 introduces the new compliant contact model and Section 5.3

presents some numerical results concerning the effects of parameters in the new model.

Finally, Section 5.4 provides concluding remarks.

⋆ Portions of the materials in this chapter are reprinted from Transactions of ASME: Journal of Applied

Mechanics, vol. 81, no. 2, Xiaogang Xiong and Ryo Kikuuwe and Motoji Yamamoto, “A Contact Force Model

With Nonlinear Compliance and Residual Indentation”, pp. 021003-1:8, 2014, as published in [97, 98], with

permission from ASME (see Appendix A2).

65

Chapter 5. A contact model with nonlinear compliance 66

5.1 Related work

In some previous work, the force-indentation curves are obtained through experiments using

real objects such as biological tissues [2, 3] and sports balls [4, 15]. One typical curve is

roughly depicted as Fig. 5.1(a), by referring to the experimental data of biological tissue

contact in [2]. It shares three distinct characteristics that has been elucidated in the beginning

of this chapter with other experimental results of real objects contact [4, 15, 60].

Kelvin-Voigt (KV) model is known to be one of the simplest models of contact. In this

model, the contact force f ∈ R can be described in the following form:

f =

0 if p < 0

−K(p + b1p) if p ≥ 0(5.1)

where p ∈ R is the displacement, of which the positive value stands for the depth of inden-

tation, K > 0 is the stiffness coefficient, and b1 ≥ 0 is the ratio of the viscous coefficient

to the stiffness coefficient. This model is preferred in many applications due to its simplic-

ity, but it produces unnatural discontinuous and sticky force as illustrated in Fig. 5.1(b) and

pointed out in [16, 53]. In addition, the linear force-indentation relation does not agree with

Hertz’s model. The COR corresponding to this model is independent from impact velocity

[51, 53].

Hunt-Crossley (HC) model overcomes the drawbacks of KV model (5.1) and is consis-

tent with Hertz’s model. HC model can be described as follows:

f =

0 if p < 0

−Kpλ(1 + b2p) if p ≥ 0,(5.2)

where b2 ≥ 0 is a damping parameter and λ > 1 is a constant related to materials and

geometries of contact objects. This model results in a force-indentation curve as illustrated

in Fig. 5.1(c) and has been utilized to reproduce the behaviors of measurably-deformable

objects [2, 3, 58–60]. However, as can be seen from the difference between Fig. 5.1(a) and

Fig. 5.1(c), HC model produces a distinctly different curve from empirical results, producing


(b)(a)

(c) (d)

0

Forc

e

Indentation

0

Forc

e

Indentation

0

Forc

e

Indentation

0

Fo

rce

Indentation

non-zero indentation

sticky force

sticky force

discontinuous

force

non-zero indentation

continuous

force

continuous

force

continuous

force

zero indentation

Figure 5.1: Force-indentation curves; (a) typical empirical result adopted from, e.g., [2,

Fig. 4], [3, Fig. 2.6] and [4, Fig. 2], (b) KV model (5.1), (c) HC model (5.2) without any

external force (solid) and with an external pulling force (dashed), (d) the author’s previous

contact model (5.3). Figure and caption are reprinted, with permission from ASME: Journal

of Applied Mechanics, “A Multistate Friction Model Described by Continuous Differen-

tial Equations”, vol. 81, no. 2, pp. 021003-1:8, 2014, Xiaogang Xiong, Ryo Kikuuwe, and



zero indentation at the time of zero contact force. It is consistent with empirical data only

when the COR is large [61]. Another weakness of HC model is that, as pointed out in [55],

an unnatural sticky force may arise when the objects are separated by an external force, as

indicated by the dashed curve in Fig. 5.1(c).

In Chapter 3, the author proposed a linear contact model defined by the following differ-

ential algebraic inclusion (DAI):

0 ∈ a +

(

1

γ+ β1

)

a − dio(γ(a − p) + a) (5.3a)

f = −K

(

a +

(

1

γ+ β1

)

a

)

. (5.3b)

Here, a ∈ R is a state variable newly introduced, β1 ≥ 0 is a constant that affects damping,

and γ > 0 is an appropriate constant. Although the DAI (5.3) does not appear usable with

common numerical integration schemes, it can be equivalently rewritten into the following

ODE by using Theorem 2:

a = max

(

−γa

1 + γβ1

, γ(p − a)

)

(5.4a)

f = −K max (0, p + β1γ(p − a)) . (5.4b)

This means that the contact model (5.3) can be used for numerical simulation employing

common ODE solvers. The behaviors of the state variable a is defined by the ODE (5.4a)

and the force f is determined by the output equation (5.4b). The Chapter 3 showed some nu-

merical examples regarding the effects of β1 and γ, but tuning guidelines for the parameters

are not obtained yet.

Equation (5.4) implies that, in the model (5.3), the force f is always nonnegative and

continuous with respect to p and a. This means that the model (5.3) never produces dis-

continuous or sticky forces, which are main drawbacks of KV model. In addition, a simple

simulation result suggests that the model produces non-zero indentation at the time of the

contact force being lost, as in Fig. 5.1(d), while HC model results in zero indentation. One

limitation of the model (5.3) is that its force-indentation curve is not close to those of Hertz’s


and HC models.

5.2 New contact model

5.2.1 Formulation of new model

To remove the limitation of linear contact-force produced by the previous contact model

(5.3), this chapter proposes a new model defined by the following DAI:

0 ∈ a +

(

1

γ+ β1 + β2a

)

a − dio(γ(a − p|p|λ−1) + a) (5.5a)

f = −K

(

a +

(

1

γ+ β1 + β2a

)

a

)

. (5.5b)

Here, λ ≥ 1 is a constant that determines the nonlinearity of the force-indentation relation

and β1 ≥ 0 and β2 ≥ 0 are damping parameters. The parameter γ > 0 is an appropriate

constant that makes it possible to rewrite the DAI (5.5) into an ODE. The units for the

quantities a, K, β1, β2 and γ are mλ, N/mλ, s, s/mλ and s−1, respectively. It is easy to see

that the previous model (5.3) is a special case of the new model (5.5) with λ = 1 and β2 = 0.

In the same manner as in the previous model (5.3), the DAI (5.3) can also be rewrit-

ten as an ODE that is tractable with numerical integration schemes. By the application of

Theorem 2, (5.5) can be rewritten as follows:

a = max

(

−γa

1 + γ(β1 + β2a), γ(p|p|λ−1 − a)

)

(5.6a)

f = −K max(

0, p|p|λ−1 + γ(β1 + β2a)(p|p|λ−1 − a))

(5.6b)

From (5.6), one can observe that, by setting β1 = 0 and β2 = 0, (5.6b) reduces to a Hertz’s

model without dissipation. This implies that the model (5.5) can also be viewed as an

extension of Hertz’s model.


Indentation (10¡2m)

Conta

ct

forc

e (

N)

new HC

Conta

ct f

orc

e (N

)

Time (10¡1s)

new HC

0.0 0.5 1.0 1.5 2.0

0

10

20

30

40

50

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

10

20

30

40

50

(a) (b)

Figure 5.2: The force-indentation curves of the new model (5.6) and HC model (5.2); The

parameters are chosen as; fe = 0 N, λ = 1.5, K = 104 N/m1.5, γ = 2 × 103 s−1, β1 =3 × 10−3 s, β2 = 0.1 s/m1.5 and b2 = 0.35 s/m. The initial conditions are set as; p(0) =−0.1 m, p(0) = 2 m/s and a(0) = 0 m1.5. Figure and caption are reprinted, with permission

from ASME: Journal of Applied Mechanics, “A Multistate Friction Model Described by

Continuous Differential Equations”, vol. 81, no. 2, pp. 021003-1:8, 2014, Xiaogang Xiong,


5.2.2 Comparison to HC model

In this section, the new model is compared to HC model through two simulations. Both

simulations are performed for a system composed of a point mass M > 0 whose position

is denoted as p ∈ R and a surface fixed at the origin. With a contact force f ∈ R and an

external force fe ∈ R, the equation of motion of the system is described as follows:

Mp = f + fe. (5.7)

Simulations are performed based on (5.7) with f being substituted by the output of the

contact models. For the time integration, the fourth-order Runge Kutta method is used with

the timestep size 1× 10−3s. The mass M is always set as M = 1 kg. In the first simulation,

the mass runs into the fixed surface with initial velocity p(0) > 0 and fe = 0 while in the

second one, the mass falls down to the fixed surface with p(0) = 0 and is subjected by the

gravity fe = Mg where g is the gravity acceleration. The parameters of both models are set

so that they produce approximately the same COR with each other for the contact in the first


0 1 2 3 4

¡3

¡2

¡1

0

1

2

Time (s)(a)

Time (s)(b)

¡p (

m)

¡p (

m/s

)¢

new HC

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

100

200

300

400

500

Conta

ct

Forc

e (

N)

Indentation (10¡2m)(c)

new

Conta

ct

Forc

e (

N)

Indentation (10¡2m)(d)

HC

new HC

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

100

200

300

400

500

0 1 2 3 4

0.1

0.2

0.3

0.4

0.5

0.0

Figure 5.3: Simulation of bouncing motion by using the new model (5.6) and HC model

(5.2). The parameters are set as; fe = Mg, λ = 1.5, K = 104 N/m1.5, γ = 2×103 s−1, β1 =1.45 × 10−3 s, β2 = 0.2 s/m1.5 and b2 = 0.2 s/m. The initial conditions are set as; p(0) =−0.5 m, p(0) = 0 m/s and a(0) = 0 m1.5. Figure and caption are reprinted, with permission





simulation and for the first bouncing in the second simulation. The KV model (5.1) and the

author’s previous model (5.4) are not included in the comparison because of their unrealistic

patterns of force-indentation curves.

Fig. 5.2 shows the first simulation result obtained with the new model and HC model.

Fig. 5.2(a) shows that the peak value of the force-time curve of HC model shifts left, com-

paring that of the new model. Fig. 5.2(b) shows that the new model results in a non-zero

indentation (approximately 0.4 × 10−2 m) when the contact force arrives in zero, while, in

HC model, zero contact force is achieved only at the time of zero indentation. The nonzero

indentation of the new model’s curve implies that the contact object lose contact before it

recover to its original shape. The new model’s curve is also characterized by its down-

ward convex shape similar to Hertz’s model. Such features of the new model are in good

agreement with empirical data in the literature, illustrated in Fig. 5.1(a).

Fig. 5.3(a) and (b) show the position and velocity results of the second simulation. It can

be seen that, although the two models produce almost the same height of the second peak

with each other, their subsequent behaviors are distinctly different. Specifically, the new

model stops bouncing in a shorter time than the HC model. This difference can be attributed

to the term Kβ1a in the new model (5.5), which dissipate energy even when the penetration

p is shallow.

Fig. 5.3(c) and (d) show the contact force-indentation curves. Fig. 5.3(c) shows that HC

model results in the zero indentation when the contact force arrives back to zero, while

Fig. 5.3(d) shows that the new model produces a non-zero indentation when the contact

force becomes zero. The later simulation result is consistent with the experiment results in

[4, 15].

5.3 Effects of parameters β1, β2 and γ

5.3.1 Force-indentation curves

In this section, the behaviors of the new model and the effects of the parameters of the new

model are illustrated by some simulation results. Those results are obtained by integrating

the system (5.7).


Time (10¡2s)(a)

Indentation (10¡2m)(b)

Fo

rce (

N)

¯1=0, 1, 2, 3, 4 £ 10¡3 s (from slender to rounded);

¯2=0 s/m1.5;

¯1=0, 1, 2, 3, 4£10¡3 s (from right to left);

¯2=0 s/m1.5;

°=1000 s¡1;

0 2 4 6 8 10 12

0

20

40

60

80

100

120

0 1 2 3 4 5

0

20

40

60

80

100

120

Fo

rce (

N)

Fo

rce (

N)

Time (10¡2s)(c)


¯1=0, 1, 2, 3, 4£10¡3 s (from right to left);

¯2=1 s/m1.5;

°=1000 s¡1;


¯2=1 s/m1.5;

°=1000 s¡1;

0 1 2 3 4 5

0

20

40

60

80

100

120

0 2 4 6 8 10 12

0

20

40

60

80

100

120


¯2=2 s/m1.5;

°=1000 s¡1;

¯1=0, 1, 2, 3, 4 £ 10¡3 s (from right to left);

¯2=2 s/m1.5;

°=1000 s¡1;

0 1 2 3 4 5

0

20

40

60

80

100

120

0 2 4 6 8 10 12

0

20

40

60

80

100

120

Time (10¡2s)(e)

Indentation (10¡2m)(f)

Fo

rce (

N)

Fo

rce (

N)

Fo

rce (

N)

°=1000 s¡1;

Figure 5.4: Influence of β1 on the behaviors of the new model (5.5); The parameters are

set as; fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The initial conditions are set as; p(0) =−0.1 m, p(0) = 2 m/s and a(0) = 0 m1.5. Figure and caption are reprinted, with permission





Time (10¡2s)

Forc

e (

N)

(a)Indentation (10¡2m)

(b)

Forc

e (

N)

¯1=0 s;

¯2=0, 1, 2, 3, 4 s/m1.5 (from

slender to rounded);

°=1000 s¡1;

¯1=0 s;

¯2=0, 1, 2, 3, 4 s/m1.5

(from right to left);

°=1000 s¡1;

Time (10¡2s)

Forc

e (

N)

(c)

¯1=2£10¡3 s;

¯2=0, 1, 2, 3, 4 s/m1.5

(from right to left);

°=1000 s¡1;


Forc

e (

N)

¯1=2£10¡3 s;

¯2=0, 1, 2, 3, 4 s/m1.5 (from


°=1000 s¡1;

Forc

e (

N)

Forc

e (

N)

Indentation (10¡2m)Time (10¡2s)(e) (f)

¯1=4£10¡3 s;

¯2=0, 1, 2, 3, 4 s/m1.5 (from

right to left);

°=1000 s¡1;

¯1=4£10¡3 s;

¯2=0, 1, 2, 3, 4 s/m1.5(from


°=1000 s¡1;

0 1 2 3 4 5

0

20

40

60

80

100

120

0 2 4 6 8 10 12

0

20

40

60

80

100

120

0 1 2 3 4 5

0

20

40

60

80

100

120

0 2 4 6 8 10 12

0

20

40

60

80

100

120

0 1 2 3 4 5

0

20

40

60

80

100

120

0 2 4 6 8 10 12

0

20

40

60

80

100

120

Figure 5.5: Influence of β2 on the behaviors of the new model (5.5); The parameters are

set as; fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The initial conditions are set as; p(0) =−0.1 m, p(0) = 2 m/s and a(0) = 0 m1.5. Figure and caption are reprinted, with permission





Time (10¡2s) Indentation (10¡2m)

(a) (b)

¯1=2£10¡3 s;

¯2=1 s/m1.5;

°=50, 100, 250, 500, 1000 s¡1

(from acute to obtuse);

¯1=2£10¡3 s;

¯2=1 s/m1.5;

°=50, 100, 250, 500,

1000 s¡1(from right to left);

Forc

e (

N)

Forc

e (

N)

¯1=4£10¡3 s;

¯2=2 s/m1.5;

°=50, 100, 250, 500,

1000 s¡1 (from acute

to obtuse);

¯1=4£10¡3 s;

¯2=2 s/m1.5;

°=50, 100, 250, 500,

1000 s¡1(from right to left);

¯1=6£10¡3 s;

¯2=3 s/m1.5;

°=50, 100, 250, 500,

1000 s¡1 (from acute

to obtuse);

¯1=6£10¡3 s;

¯2=3 s/m1.5;

°=50, 100, 250, 500,

1000 s¡1(from right to

left);

Forc

e (

N)

Forc

e (

N)

Forc

e (

N)

Forc

e (

N)

Time (10¡2s)

(c)


(d)

Time (10¡2s)

(e)


(f)

0 2 4 6 8 10 12

0

20

40

60

80

100

120

0 1 2 3 4 5

0

20

40

60

80

100

120

0 2 4 6 8 10 12

0

20

40

60

80

100

120

0 1 2 3 4 5

0

20

40

60

80

100

120

0 2 4 6 8 10 12

0

20

40

60

80

100

120

0 1 2 3 4 5

0

20

40

60

80

100

120

Figure 5.6: Influence of γ on the behaviors of the new model (5.5); The parameters are set

as; fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The initial conditions are set as; p(0) = −0.1 m

and a(0) = 0 m1.5. Figure and caption are reprinted, with permission from ASME: Journal





First, the effects of the three parameters β1, β2 and γ are numerically investigated.

Fig. 5.4 shows the effect of β1 with β2 fixed at different values. For each fixed β2, the varia-

tion of β1 shows that the residual indentation increases as β1 increases, as seen in Fig. 5.4(b),

(d) or (f). In the special case of β1 = 0 in Fig. 5.4(d) and (f), the force-indentation curve

is close to that of HC model. Another important effect is that, as is seen in Fig. 5.4 (a), (c)

or (e), an increase of β1 results in an increase of the rate-of-change of the contact force (the

slope of the curves in Fig. 5.4(a), (c) or (e)) at the beginning of contact. With a non-zero

value of β1, the force rate-of-change discontinuously changes with respect to time at the

time of collision.

Next, Fig. 5.5 shows the effect of β2 with β1 fixed at different values. For each fixed

β1, the variation of β2 shows that the force-indentation curve becomes more rounded as β2

increases, as illustrated in Fig. 5.5(b), (d) or (f). In the special case of β1 = 0, the force-

indentation curve is close to that of HC model, as illustrated in Fig. 5.5(b). In addition, a

larger β2 results in the leftward shift of the peak time of the peak contact force and shorter

duration of the compression phase. It should be noted that β2 does not influence the residual

indentation at the time of loss of contact force or the rate-of-change of the force at the time

of collision.

Fig. 5.6 shows the effect of γ with β1 and β2 being fixed. The overall shape of the

contact force-indentation is influenced by the value of γ. Fig. 5.6(a), (c) and (e) show that,

as γ increases, the force-time curves become more unsymmetrical and the duration of the

compression phase becomes shorter. Fig. 5.6(b), (d) and (f) show that, as γ becomes larger,

the shape of the contact force-indentation changes from cute, triangle-like curves as those

produced by the models in [49, 61–64] to obtuse, rounded curves as those produced by HC

model.

From Fig. 5.4, Fig. 5.5 and Fig. 5.6, one may conclude that β1 determines the residual

indentation, β2 determines the roundedness of the contact force-indentation curves, and that

γ determines the overall shape of the contact force-indentation curves.


0 1 2 3 4 50

2

4

6

8

0 1 2 3 4 50

2

4

6

8

1

0

¯1=4£10¡3 s; ¯2=2 s/m1.5;

¯2 (

s/m

1.5)

1

0

¯1 (

10¡

3s)

Impact velocity (m/s)

(a)


(b)

¯1=0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5£10¡3 s

(from top to bottom) ¯2=0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5 s/m1.5

(from top to bottom)

¯1=4£10¡3 s; ¯2=2 s/m1.5;

CO

R

CO

R

(d)Impact velocity (m/s)

(c) Impact velocity (m/s)

1 2 3 4 50.0

0.2

0.4

0.6

1.0

0.8

0 0

°=1000 s¡1; °=1000 s¡1;

°=1000 s¡1; °=1000 s¡1;

1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Figure 5.7: (a)(b) The COR as a function of β1, β2 and the impact velocity obtained from

of the new model (5.5). (c)(d) The COR as a function of impact velocity. The parameters

are set as; fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The initial conditions are set as;

p(0) = −0.1 m and a(0) = 0 m1.5. Figure and caption are reprinted, with permission





0 1 2 3 4 50.0

0.5

1.0

1.5

2.0

2.5


(a)


(b)

CO

R

°=25, 75, 125, 175, 250, 375,

500, 1000, 1750, 2500 s¡1,

(from top to bottom)

¯1=4£10¡3 s;

¯2=2 s/m1.5;

1

0

¯1=4£10¡3 s;

¯2=2 s/m1.5;

° (

10 3

s¡

1)

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

Figure 5.8: (a) The COR as a function of γ and the impact velocity obtained from of the

new model (5.5). (b) The COR as a function of impact velocity. The parameters are set as;

fe = 0 N, λ = 1.5 and K = 104 N/m1.5. The initial conditions are set as; p(0) = −0.1 m

and a(0) = 0 m1.5. Figure and caption are reprinted, with permission from ASME: Journal





5.3.2 Coefficient of restitution (COR)

From empirical studies in the literature [53, 60], it has been known that the COR decreases

as the impact velocity increases. A set of simulation was performed to investigate the COR

obtained by the new model.

Fig. 5.7(a) and (b) show the result, in which the COR is shown as a function of the impact

velocity and the parameters β1 and β2. Fig. 5.7(c) and (d) show partial data sets from the

data in Fig. 5.7(a) and (b), respectively. In all graphs, it can be seen that COR decreases

as the impact velocity increases with any settings of β1 and β2. This feature is consistent

with the known fact in the literature [53]. In addition, Fig. 5.7(c) and Fig. 5.7(d) both show

that COR decreases as β1 or β2 increases, which is as expected considering that they are

damping factors.

An important point in Fig. 5.7(c) and (d) is that, as the impact velocity approaches to

zero, the COR does not converges to 1 except the special case β1 = 0 shown in Fig. 5.7(c).

This observation is consistent with the fact that, with β1 = 0, the new model becomes close

to HC model as illustrated in Fig. 5.5(b), and that HC model is intended to realize COR= 1

with the extreme case of zero impact velocity [16]. Another important point is that, in

another special case of β2 = 0, the corresponding COR is almost independent from the

impact velocity, as shown by the top curve with a slight slope in Fig. 5.7(d). This supports

the necessity of the term Kβ2aa with β2 > 0 in the new model (5.5) to reproduce the COR

depending on the impact velocity, which has been empirically known [51, 52, 60].

Fig. 5.8(a) shows the COR as a function of impact velocity and the parameter γ. The

data of Fig. 5.8(b) is a part of those in Fig. 5.8(a). Fig. 5.8(b) shows that the monotonic

decrease of COR with respect to the impact velocity is preserved at any values of γ, except

the small value of γ, shown by the top curve in Fig. 5.8(b) and conformed by the bottom part

of Fig. 5.8(a). This exception shows that when γ is small, the COR increases with a slight

slope with respect to the impact velocity and this gives an agreement with the experiment

result for the contact of a baseball dropped from different heights [15], which shows that the

COR increases as the height becomes large. Fig. 5.8(b) also shows that, with fixed impact

velocity, the COR decreases as γ increases, although it becomes very insensitive to γ when

γ is large enough, illustrated by the almost parallel curves of top part of Fig. 5.8(a) and


the covered curves in Fig. 5.8(b). This fact is in contrast to the fact shown in Fig. 5.6(d),

in which there are slight variation in the contact force-indentation curves even when γ is

large. This means that a variation in γ has an effect of changing the shape of the contact

force-indentation curve without causing much variation in the amount of energy loss.

In conclusions, one can see different effects of β1, β2 and γ on the COR-velocity curve

from Fig. 5.7(c), (d) and Fig. 5.8(b). The parameter β1 determines the COR at the zero

impact velocity. The parameter β2 determines the slope and the overall shape of the curve.

The parameter γ influences both the zero-velocity COR and the slope, but its influence

becomes smaller as its value increases.

5.4 Summary

This chapter has proposed an extension of the author’ previous compliant contact model

for the consistency with empirical results in the literature. The proposed model produces

the three characteristics shown in the very beginning of this chapter. The proposed model

contains two parameters β1, β2 for damping, two parameters K,λ for static stiffness

and one parameter γ for shapes of the contact-force curves. Numerical results have shown

that β1 determines the magnitude of residual indentation and the rate-of-change of the force

immediately after the collision, that β2 influences the roundedness of the hysteresis curve in

the force-indentation plane, and that γ influences the shapes of the contact force-indentation

curves. The new model also reproduce a negative correlation between the COR and the

impact velocity, which is a known result in the literature.

Future work should address design guidelines for the parameters to produce intended

shapes of hysteresis curves and COR. Theoretical investigation on the properties of DAI

(5.5) would be necessary to clarify the effects of various factors, such as the stiffness pa-

rameters K and λ, the damping factors β1 and β2 and the coefficient γ. In addition, a further

extension of the model to include friction and oblique collision should be sought as a future

topic of study.

Chapter 6

Concluding remarks

This dissertation focuses on the modeling of friction and contact by using differential alge-

braic inclusions (DAIs). It starts from the concept of differential inclusions (DIs), which are

set-valued generalizations of ordinary differential equations (ODEs). Mechanical systems

involving friction and contact are described as DIs when the contact bodies are idealized as

rigid ones and impenetrable to each other. Integrations of DIs are troublesome due to the

DIs’ set-valued characteristics. In conventional regularization approaches, DIs are directly

approximated as ODEs for the easiness of numerical integration. Those straightforward

approximations lack the discontinuous nature of original DIs and can cause problems of

unnatural behaviors. In this dissertation, DIs are first regularized as DAIs that inherit the

discontinuous nature of original DIs. Then the DAIs provide a single-state friction model

and a linear contact model to approximate DIs as ODEs. To enhance the applicability of

friction and contact models in various engineering applications, the single-state friction and

linear contact models are extended to more sophisticated versions such that they can capture

more features of friction and contact phenomena. The single-state friction model is extend

to a multistate version. It can capture the major friction properties that previous friction

models do, such as the Stribeck effect, nondrifting property, stick-slip oscillation, presliding

hysteresis with nonlocal memory, and frictional lag. Moreover, it is free from the problems

that previous models suffer from, such as unbounded positional drift or discontinuous force.

The linear contact model is extended to a nonlinear version. It can simultaneously satisfy the

81

Chapter 6. Concluding remarks 82

major features of experimental data of soft-objects contact, such as continuity of the force at

the time of collision, Hertz-like nonlinear force-indentation curve, and non-zero indentation

at the time of loss of contact force. In contrast, previous contact models can only capture

one or two of the three features.

The dissertation has first provided an overview of the conventional approaches to de-

scribe mechanical systems involving friction and contact phenomena. Those conventional

approaches can be broadly categorized into two classes: hard-constraint approaches and

regularization approaches. The hard-constraint approaches typically describe mechanical

systems as DIs. The DIs are cumbersome to be integrated due to their set-valued charac-

teristics. Regularization approaches can be used to approximate those DIs by ODEs and

then lead to simple integration procedures. One of major problems of the regularization ap-

proaches is that they usually lack the discontinuous nature of the original DIs, which cause

various unnatural behaviors in descriptions of mechanical systems.

To solve the above problems of hard-constraint and regularization approaches, Chapter 3

has proposed a new approach to regularize DIs. Different from conventional regularization

approaches, here, DIs have been first relaxed into DAIs that can inherit the discontinuous

nature of DIs. Then, the DAIs were transformed into ODEs, providing a single-state friction

model and a linear contact model. This method provides a simple procedure to effectively

simulate mechanical systems involving friction and contact with preserving the discontinu-

ous nature. Three examples have been taken to illustrate the simplicity and efficiency of the

proposed method. Those examples have shown that, excepting some small penetrations, the

proposed method appropriately describe the overall behaviors of mechanical systems that

are originally described by DIs.

To extend the application scope of previous friction and contact models to engineering

applications, Chapter 4 and Chapter 5 have extended the single-state friction model and

linear contact model in Chapter 3 into more sophisticated versions such that they can capture

more features of experimental results of friction and contact phenomenon and inherit their

advantages, i.e., preserving the discontinuous nature.

Chapter 4 has extended the single-state friction model in Chapter 3 into a multistate

version. This extended model is composed of parallel connection of a number of elastoplas-

Chapter 6. Concluding remarks 83

tic elements. Each elastoplastic element, representing an asperity on the contact surfaces

involving friction, has been described by the single-state friction model. This new model

reproduces major features of friction phenomena reported in the literature [14, 21, 35, 38],

such as the Stribeck effect, nondrifting property, stick-slip oscillation, presliding hystere-

sis with nonlocal memory, and frictional lag. Moreover, the new model does suffer from

the problems of previous regularized models. The new model has been validated through

comparisons between its simulation results and empirical results reported in the literature

[14, 22]. It also has been experimentally validated in [99].

Chapter 5 has extended the linear contact model in Chapter 3 to a nonlinear version.

The new contact model has been equipped with a Hertz-like power-law nonlinearity and

a displacement dependent viscosity term based on its original version. The contact force

and indentation of this model is consistent with the experimental data in the following three

features: (i) continuity of the force at the time of collision, (ii) Hertz-like nonlinear force-

indentation curve, and (iii) non-zero indentation at the time of loss of contact force. On the

contrary, the conventional KV model only satisfies (ii) while HC model only satisfies (i) and

(ii). The new model has been validated through comparisons among the simulation results

of the new model, the HC model, and empirical results reported in the literature [2–4]. The

parameter effects and the COR properties of the new model have been investigated through

numerical simulations.

This dissertation has some limitations on its incomplete contents. Although the mul-

tistate friction model has been validated by experiments in [99], its parameters were only

identified by try and error, which is a rough way and can not achieve optimal performance

of friction compensation. Real-time methods for identifying the multiple parameters of the

multistate friction model should be developed for the easinesses of engineering applications.

Another limitation is that the performances of the proposed nonlinear contact model have

only been validated through numerical methods. Experimental validations should be done

in future study. Meanwhile, the guideline of parameter adjustments for this contact model

should be illustrated through simulations and experiments.

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Appendix A: Copyright Permissions

Appendix A1: Copyright permission for Chapter 4

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Appendix A2: Copyright permission for Chapter 5

Appendix 100

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